Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics Olariu E.andFlorentin Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics February, 2017 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics - Lecture 1 The Structure of the Course, Scores and Grades Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics Probabilities and Statistics First seven weeks areProbabilities dedicated to the Theory of ProbabilProbabilities and Statistics Probabilities and Statistics Probabilities and Statistics ity (courses and seminars), the last Probabilities six concern the Probabilities and Statistics Probabilitieswhile and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Statistics (courses and laboratories). Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Probabilities andseven Statisticslaboratories Probabilities and Statistics We and willStatistics have six seminars, and thirteen Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics courses. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Partial exam from the first seven courses in the 8-th week Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics (score T - optional. Those who fail Probabilities to receive at Probabilities and1 )Statistics Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics least 30 (from a maximum of 60) points must retake Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics Probabilities Statisticsin the Probabilities and Statistics the and Probability Theoryandexam arrears session Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics (T 30). This exam can be taken also in the regular Probabilities1 and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics andweek). Statistics Probabilities and Statistics sessionand(instead of Probabilities the 8-th Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Statistics exam from Probabilities the last and sixStatistics courses inProbabilities the regular sesProbabilities and Statistics and Statistics Probabilities and StatisticsTS ). Probabilities and Statistics Probabilities andat Statistics sion (score Those who fail to receive least Probabilities and Statistics2 Probabilities and Statistics Probabilities and Statistics 30 (from a maximum points Probabilities must retake the Probabilities and Statistics Probabilitiesof and60) Statistics and Statistics > Statistics exam in the arrears session (T2 > 30). The Structure of the Course, Scores and Grades Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics From seminars and labs you will get anotherProbabilities two intermmeProbabilities and Statistics Probabilities and Statistics and Statistics Probabilities StatisticsT3 and Probabilities Probabilities and Statistics diateandscores T4 : and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics T3 comes from sixProbabilities small tests one on each seminar. Those Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics who fail to receive at least 30 (from a maximum of Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 6 10 and60) points Probabilities cannot and pass the course andandmust Probabilities Statistics Statistics Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics retake the course in the next year (T3Probabilities 30). and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics T4 comes partially from the presence and activity in class Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics (20points) and partially from homeworksProbabilities (40 points) whose Probabilities and Statistics Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities Probabilities Statistics end-terms are in the last weekandofStatistics the semester. Thisandgrade Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics also must be at least 30 (from a maximum of 20and Statistics 40 Probabilities and Statistics Probabilities and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 60) points. Those who fail to receive 30 points cannot Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics passandthe courseProbabilities and must retake the course in the Probabilities Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics next year (T 30). Probabilities and Statistics 4 Probabilities and Statistics Probabilities and Statistics = > + = > The Structure of the Course, Scores and Grades Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics Probabilities and Statistics Theand final score Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics T1 Tand T3 T4 Probabilities and Statistics Probabilities and Statistics T Probabilities 2 Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Thisandscore will beProbabilities graded and onStatistics the normal (Gauss) Probabilities Statistics Probabilities and curve Statistics in Probabilities Statistics Probabilities and Statistics order toandget the true Probabilities grade. and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics All intermediate scores – Tand T2 T3 , andProbabilities T4 – must be Probabilities and Statistics Probabilities and Statistics 1 Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics at least 30, otherwise the exam will not be considered Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics passed. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics If Tand T4 is not at least 30, then Probabilities you cannot take Probabilities Statistics Probabilities and Statistics and Statistics 3 or Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics the exams in the regular or arrears session. If T1 or Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics T2 is not at least 30, you cannot course. Probabilities and Statistics Probabilities and Statisticspass the Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics In the and arrears all the scores must be at Probabilities Statistics session Probabilities and Statistics Probabilities andleast Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 30. = + + + ; ; : Table of contents Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and experience Statistics Probabilities and Statistics 1Probabilities Random and random events. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Random experience Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Random events Probabilities Probabilities and Statistics and Statistics Probabilities and Statistics Random notations Probabilities and Statistics Probabilities andevents Statistics - properties Probabilitiesand and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 2Probabilities Probability functionProbabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probability function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probability of elementary random events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 3 Exercises Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Random Probabilities andEvents Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 4 Bibliography Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Random experience Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Intuitively, a randomProbabilities experience corresponds to a and process Probabilities and Statistics and Statistics Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics producing a result which is not known before performing Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics whatProbabilities andisStatistics and Statistics this and process; is known the set of Probabilities all possible results. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Theand outcome an experience neutral Probabilities Statistics of such Probabilities and Statisticsare registred Probabilitiesby andaStatistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics observer. Usually, random experiences are practical ones, Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics but canand beStatistics also abstract. Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics The outcome of a random experience are dueProbabilities to chance only. Probabilities and Statistics Probabilities and Statistics and Statistics Probabilities Statistics Probabilities Statistics onlyProbabilities and Statistics At aand given moment we willand consider one random eperiProbabilities and Statistics Probabilities and Statistics Probabilities and Statistics enceand (even if this Probabilities experience be re-performed). Probabilities Statistics and can Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Random experience Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Probabilities and Statistics Probabilities and Statistics Definition 1Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics A random experience is an experience resultandisStatistics not Probabilities and Statistics Probabilities and Statistics whose Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics known before peforming it, but those all possible results are Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics known, and which can be performed same conditions Probabilities and Statistics Probabilities and Statistics in the Probabilities and Statistics Probabilities andtime. Statistics Probabilities and Statistics Probabilities and Statistics at any given Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Examples: Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics By tossing a die the possible 1 2 6 ,Statistics but Probabilities and Statistics Probabilitiesoutcomes and Statistics are Probabilities and Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics we doesn’t know for sure which number will appear. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities By tossing two dice the possible outcomes are: and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 1 1Probabilities 1 2 and Statistics 6 6 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statisticsare Probabilities By flipping two coins, the outcomes H H , and H Statistics T , Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics T H ,andTStatistics T . Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics If the experience is to measure the lifespan of a battery the f ; ;:::; g | f( ; ); ( ; ); : : : ; ( ; )g: | f( ; ) ( ; ) ( ; ) ( ; )g | outcome will be a real non-negative number. | Elementary random events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics and Statistics Following a random Probabilities experience means to Probabilities observe its outProbabilities and Statistics Probabilities and Statistics Probabilities and Statistics come; each time we will have only one outcome (from the Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics known set of all possible results). We say in this case that Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics an elementary event is produced. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Definition 2Statistics Probabilities and Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Statistics andelemenStatistics A possible result of a Probabilities random and experience is Probabilities called an Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics tary random event, Probabilities the set of these events is the sample Probabilities and Statistics andall Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics space, denoted by . Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics After an experience only one elementary event Probabilities andperforming Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics Statistics Probabilities and Statistics from and "happens" - Probabilities followingandthe "chances" each outcome Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics has. and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics In the process of making reasoning about the experience, our Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Statistics itselfProbabilities interest can be notProbabilities for the and outcome but for and theStatistics family to which it belongs. Random events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics The above observation extendsandthe notion ofProbabilities elementary ranProbabilities and Statistics Probabilities Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics dom event. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Definition 3Statistics Probabilities and Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics and Statistics A random event is Probabilities a certain subset of Probabilities the sample space: Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics A . Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics We and tossStatistics a die: theProbabilities sampleand space is 1 2 and6Statistics ; if we Probabilities Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics are interested in even numbers, the corresponding random Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics event isand AStatistics 2 4 6 .Probabilities and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics We toss two dice: 1 1 1 2 6 6 ; if we are Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics interested in a sum equal with 4, the corresponding random Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics event is A 1 3 2 2 3 1 . Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and two Statistics and Statistics If we flip coins: Probabilities H andHStatistics H T Probabilities T H T T . Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics The random event: Probabilities "the twoandsides is A Probabilities and Statistics Statisticsare different” Probabilities and Statistics Probabilities and Statistics Probabilities Statistics T and HStatistics T H .Probabilities The event "at least one of theandtwo sides =f =f ; ; is head ” iss B g g | = f( ; ); ( ; ); : : : ; ( ; )g = f( ; ); ( ; ); ( ; )g | = f( ; ); ( f( ; ); ( ; )g ; ;:::; ; ); ( ; = f(H ; H ); (H ; T ); (T ; H )g. | ); ( ; )g = Random event Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics Usually a randomProbabilities event isanddefined using aProbabilities predicate: the Probabilities and Statistics Statistics Probabilities and Statistics Probabilities and Statistics and Statistics Statistics event will be formedProbabilities with the elementaryProbabilities randomandevents Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics which make the predicate true. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities andaStatistics Probabilities and space Statistics . Formally, a random event is subset of the sample Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 0 ), Probabilities Statistics of If and is Statistics a discrete Probabilities set (i. e., then anyandsubset Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics A and Statistics can be considered random Probabilities Probabilitiesaand Statistics event. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics If and is Statistics not discrete, then only specific subsets can be Probabilities Probabilities and Statistics Probabilitiesofand Statistics Probabilities Statistics- this Probabilities andsubject Statistics of our Probabilities and Statistics randomandevents is not the course. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics j j 6 @ Random events - properties and notations Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Definition 4 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics We say that a random event and A Statistics occursProbabilities if, afterandperformProbabilities and Statistics Probabilities Statistics and Statistics Probabilities and Statistics random Probabilities and Statistics ingProbabilities the random experience, the elementary event (the Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics outcome) to A. Probabilities and Statistics Probabilitiesbelongs and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities We denote random events using big caps like A, B ,and CStatistics etc. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Because in our applications will be a discrete set, we will Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities consider that any Probabilities subset ofand Statistics is a random event.and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities andnow Statistics Probabilities and Statistics Probabilities and Statistics From on we do not make any distinction between subProbabilities and Statistics Probabilities and Statistics Probabilities and Statistics sets and of Statistics and random events; operations Probabilities and properties over Probabilities Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics subsets are transfered on random events. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics Probabilities and(it Statistics isand the impossible event will neverProbabilities occur);and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics isand total event (it occursandeverytime); Probabilities Statistics Probabilities Statistics Probabilities and Statistics ? Random events - properties and notations Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics - if A and B Statisticsare random events, then Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and A[ B is a random eventandwhich occurs Probabilities if and only one of Probabilities Statistics Probabilities Statistics andifStatistics Probabilities Statistics Probabilities and Statistics theand events A or BProbabilities occurs; and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics A \ B is a randomProbabilities event which occurs if and only if both Probabilities and Statistics and Statistics Probabilities and Statistics events A and B occurs; and Statistics Probabilities and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities A nand B Statistics is a randomProbabilities event which occurs if and only ifand A Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics occurs but B doesn’t; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics -Probabilities if A is and a random then and A Statistics A is Probabilities a random Statistics event, Probabilities andevent Statistics Probabilities and Statistics Probabilitiesof andA: Statistics Probabilities Statistics called the complement A occurs if andandonly if A Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics doesn’t occur; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics if A B , we say that A implies B ; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Statistics Probabilities andB Statistics -Probabilities if A Band Statistics we say Probabilities that A isandnot compatible with (or Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics disjoint of B ); we also say that the events are mutually Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ; = n \ =? exclusive); if A \ B 6= ?, then A is compatible with B . Random events - properties and notations Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics manner Probabilities and Statistics and Statistics In aand similar we define union orProbabilities intersection of a Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics finite (or infinitely countable) number of events. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Example. the tossingand ofStatistics two dice: Probabilities and Statistics Probabilities andConsider Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities andiStatistics Probabilities and Statistics Probabilities and Statistics j 1 i j 6 there are 36 random events. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics andevent Statistics“the sum Probabilities Statistics Probabilities and Statistics LetProbabilities A be the of theanddice is 4” and B “the dice Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics are Probabilities greater than 4”: and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics AProbabilities 1 3and Statistics 2 2 3 1 Probabilities and B and Statistics 5 5 5 6Probabilities 6 5 and 6 6Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics -Probabilities A B and Statistics 1 3 2 2 Probabilities 3 1 and 5 5Statistics 5 6 6 Probabilities 5 6 6 and; Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics -Probabilities A B and Statistics - A and B are disjoint; Probabilities and Statistics Probabilities and Statistics Probabilities Probabilities - A and Statistics 1 1 1 2 Probabilities 2 1 3 and 2 Statistics 2 3 6 6 . and Statistics = f( ; ) : 6 ; 6 g; = = f( ; ); ( ; ); ( ; )g = f( ; ); ( ; ); ( ; ); ( ; )g [ = f( ; ); ( ; ); ( ; ); ( ; ); ( ; ); ( ; ); ( ; )g \ =? = f( ; ); ( ; ); ( ; ); ( ; ); ( ; ); : : : ( ; )g | Probability function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statisticsin this Probabilities Statistics Probabilities and Statistics We consider section and that is at most infinitely countProbabilities and Statistics Probabilities and Statistics Probabilities and Statistics able:and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 1 2 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics The notion of probability comes from theProbabilities idea of associating Probabilities and Statistics Probabilities and Statistics and Statistics Probabilities and Statisticsto each Probabilities and Statistics Probabilities and Statistics a real number random event in order to compare Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics their chances. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics is based Probabilities and Statistics Theand definition on and theStatistics empirical Probabilities observation that an Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics event more often must have a greater probaProbabilities andwhich Statistics occurs Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics bility than one which occurs rarely. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics of a Probabilities Statistics Probabilities The probability random and event A is denoted byandPStatistics A. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics The following axioms define the probability function. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics On the and same set weProbabilities can define different probability Probabilities Statistics and Statistics Probabilities and funcStatistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics tions! = f! ; ! ; : : :g: ( ) Axioms of probability function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Probabilities and Statistics Probabilities and Statistics Definition 5Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics (Kolmogorov) If is the sampleandspace experiProbabilities and Statistics Probabilities Statisticsof a random Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ence, a probability function on is a function P defined on Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics the random events family (inand our case ) whichandsatisfies Probabilities and Statistics Probabilities Statistics Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Axiom 1. 0 P Probabilities A 1, and forStatistics any random event Probabilities and Statistics Probabilities andA. Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Axiom 2. P 1 and and P Statistics 0. Probabilities and Statistics Probabilities and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Axiom 3. If events A1 A are mutually exProbabilities and Statistics Probabilities and2 StatisticsAk Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics clusive, then Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ! and Statistics Probabilities and Statistics Probabilities Probabilities and Statistics [ X Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics P Probabilities Ak and Statistics P Ak Probabilities and Statistics Probabilities and Statistics k 1 and Statistics k 1 Probabilities and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 6 ( )6 ( ) = ; 1 = P( ) (?) = ;:::; = 1 = ;::: ( ): Axioms of probability function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statisticsasociated Probabilities Axiom 1 says thatProbabilities the probability withanda Statistics random Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics event a real number from 1. Probabilities andis Statistics Probabilities and 0 Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Axiom 2 says that any outcome of the experience Probabilities and Statistics Probabilities and Statistics Probabilities and belongs, Statistics Probabilities and Statistics 1, toProbabilities and Statistics Statistics with probability . Moreover, it says Probabilities that theand imposProbabilities and Statistics Probabilities and Statistics Probabilities and Statistics sible event doesn’t occur. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Axiom 3 says that for a mutually incompatible sequence of Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics random events, the probability of its union is the and sum of the Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics probabilities events. Probabilities and Statistics of these Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics If the sequence is infinite, Axiom 3 guarantees the converProbabilities and Statistics Probabilities and Statistics Probabilities and Statistics gence ofandthe corresponding series from the right member. Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics [; ] Properties of the probability function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Proposition 1 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics LetProbabilities A1 A2 and Statistics Am be a finite sequence of mutually exclusive Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics random events, then Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics !and Statistics Probabilities and Statistics Probabilities Probabilities and Statistics m m [ X Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Ak and Statistics P Ak Probabilities and Statistics Probabilities and Statistics P Probabilities k 1Statistics Probabilities and Statistics k 1Probabilities and Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics proof: Consider 3, Aand , k Probabilities m 1. and Statistics Probabilities and Statistics in Axiom Probabilities k Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Proposition 2 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Let A and B be two random events, Probabilities and Statistics Probabilities and Statisticsthen Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics (i) P Aand Statistics P A B Probabilities P A andBStatistics ; Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics (ii) P Aand Statistics B P A Probabilities P A andBStatisticsP B ;Probabilities and Statistics Probabilities Probabilities and Statistics and Statistics Probabilities and Statistics (iii) P A 1 P A Probabilities ; Probabilities Probabilities and Statistics and Statistics Probabilities and Statistics ; ;:::; = = = ( ): =? 8 > + (iv) ( )= ( \ )+ ( n ) ( [ )= ( ) ( \ )+ ( ) ( )= ( ) if A implies B (A B ), then P (A) 6 P (B ) . Properties of the probability function = = \ Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics proof: (i): andinStatistics Proposition 1 we and can take m Probabilities 2 A1 and Statistics A Probabilities Probabilities Statistics Probabilities Probabilities Statistics Probabilities and Statistics B A2 and A Statistics B . (ii): we use (i)andand Proposition 1. (iii): we can Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics use Proposition 1 and Axiom 2: 1 P P A P A . (iv): Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Statistics Statistics useProbabilities (i) and and Axiom 1 and Probabilities get P Band Statistics P B AProbabilities P B andA Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics P A P Band Statistics A P A Probabilities . Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Proposition 3 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics (Inclusion-exclusion A 1 A2 m are random Probabilities and Statistics principle) ProbabilitiesIfandAStatistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics events, then Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ! Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics m m [ and Statistics X Probabilities Probabilities X and Statistics Probabilities and Statistics P AStatistics PProbabilities Ak Ak2 and Statistics k k1 Probabilities and and Statistics P AProbabilities Probabilities Probabilities Probabilities and Statistics k 1 and Statistics k 1 1 k1and <k2Statistics m Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics X and Statistics Probabilities Probabilities and Statistics Probabilities p 1and Statistics 1 and Statistics P and AStatistics Ak2 Probabilities Akp and Statistics k1 Probabilities Probabilities 1 Statistics k1 <k2 <:::<kp Probabilities m Probabilities and and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics m 1 1 P A1 A2 Am ; = n : = ( ) = ( )+ ( ) ( ) = ( \ )+ ( n ) = ( )+ ( n )> ( ) ; = = +( ) + 6 = +( ) ( ) + 6 ; 6 6 \ ( \ ;:::; ( \ ) + + \ ::: \ + + \ ::: \ ): Properties of the probability function = Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics dem.: We use induction Probabilities on m: for property 2, Probabilities and Statistics and m Statistics2 we have Probabilities and Statistics Probabilities and Statistics and Statistics Probabilities and Statistics (ii). Suppose now thatProbabilities the above relation holds for any m 1 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics random events. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ! " and Statistics ! # Probabilities!and Statistics Probabilities Probabilities and Statistics m m[1 m[1 [ Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics P Probabilities Ak and Statistics P AkProbabilities P and Statistics Ak AProbabilities P and Am m Statistics Probabilities Probabilities and Statistics k 1 and Statistics k 1Probabilities and Statistics k 1 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics (1) !Probabilities " and Statistics #Probabilities and Statistics Probabilities and Statistics m[1 m[1 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics P and Statistics Ak Am Probabilities P Amand Statistics Probabilities andPStatistics Ak Probabilities k 1 k 1 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics WeProbabilities use the and inductive hypothesis for the first two terms: Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ! m[1 m X1 XStatistics Probabilities and Statistics Probabilities and Probabilities and Statistics P AkStatistics P Probabilities Ak P Probabilities Ak1 Aand Probabilities and and Statistics k2 Statistics Probabilities and Statistics Probabilities and Probabilities and Statistics k 1 k 1 1 k1 < k2Statistics m 1 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics (2) X Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics p 1 Probabilities ProbabilitiesPandA Statistics Statistics 1 and Statistics Ak2 Probabilities Akand k1 p = = = = = = = +( ) + = 6 = = ( ) 6 1 k1 <k1 <:::<kp m 1 6 ( \ 6 \ ( ) + ( )= \ ) + ( ): ( \ ) + + \ ::: \ + + Properties of the probability function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics mProbabilities and Statistics 1 P Probabilities A1 A2 and StatisticsAm 1 Probabilities and Statistics Probabilities and Statistics " # and Statistics Probabilities and Statistics Probabilities Probabilities and Statistics m[1 m 1 XStatistics Probabilities and Statistics Probabilities and Probabilities and Statistics P AkProbabilities Am and Statistics P Ak Probabilities Am Probabilities and Statistics and Statistics(3) k 1 k 1Statistics Probabilities and Statistics Probabilities and Probabilities and Statistics Probabilities and Statistics X Probabilities and Statistics Probabilities and Statistics P Ak1 and AkStatistics Am Probabilities and Statistics Probabilities Probabilities and Statistics 2 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 1 k1 <k2 m 1 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics X Probabilities Probabilities and Statistics Probabilities and Statistics p and 1 Statistics Akp Aand A 1 P A m Statistics k k 2 1 Statistics Probabilities and Statistics Probabilities and Probabilities k1 <k1 <:::<kpProbabilities m 1 and Statistics Probabilities and1 Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics m 1 1 PProbabilities A1 Aand Am and Statistics Probabilities and Statistics Probabilities 2 Statistics Am 1 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Combining (1), (2) andProbabilities (3) we and getStatistics the desired relation. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics +( ) ( \ = +( ) + 6 = ( \ 6 +( ) + ) = ( 6 6 ( \ ); \ ::: \ \ = ( \ \ \ \ ::: \ ) ) + + \ ::: \ \ \ ): + + Probability of elementary random events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities andofStatistics Probabilities and Statistics Let P be the probability the elementary random event Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics , . From Axiom 3, for every random event A, we Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics have and Statistics X Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics P A P (4) Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics A Probabilities and Statistics Probabilities and ! Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics If Aandcontains number of elements, then and the above Probabilities Statistics a finite Probabilities and Statistics Probabilities Statistics Probabilities Statistics Probabilitiesaand Statistics Probabilities and Statistics sum is and finite; if A contains countable infinite number of Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics elements, then the sum is an infinite series.Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics The sum is zero if A Probabilities is the impossible event. Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics We get an equivalent definition if we replace Axiom 3 with Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Axiom 3’. For every random event A, Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics X Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics P (A) = P (! ): Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics !2A Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics (!) f!g 8 ! 2 ( )= 2 (!): Equiprobable elementary random events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics In many situationsProbabilities consists of elementary random events Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics having the same probability: Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 1 Probabilities Statistics p Probabilities 1 i and n Statistics 2 nand P i Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics then and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics n Probabilities and StatisticsX Probabilities and Statistics Probabilities1and Statistics Pand Statistics 1 P i and npStatistics i. e., P Probabilities p and Statistics i Probabilities Probabilities i Probabilities and Statisticsi 1 Probabilities and Statistics Probabilitiesnand Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics in this Probabilities andrandom Statistics event Probabilities and Statistics Let us consider, case, a A, with Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics then A kand , AStatistics i1 i2Probabilitiesik and, Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics k X A Probabilities and Statistics Probabilities and Statistics k Probabilities and Statistics P A P ij Statistics Probabilities and Statistics Probabilities and Probabilities and Statistics n j 1 and Statistics Probabilities and Statistics Probabilities Probabilities and Statistics = f! ; ! ; : : : ; ! g; (! ) = ; 8 6 6 ( ) = = j j= = = f! (! ) = ; ;! ;:::;! g ( )= = ; (! ) = = ; 8 : (! ) = = jj jj : Equiprobable elementary random events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics We just proved Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Proposition 4 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics If the elementary random events are equiprobable, then the Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities of and a Statistics Probabilities andthe Statistics Probabilities andnumStatistics probability random event A is ratio between the Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics berProbabilities of elements from A and the number of elements from . and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics the Probabilities and Statistics Probabilities In this and situation probability of a random eventandisStatistics the Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics number of "favorable" cases over the total number of cases. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statisticsprobabilities Probabilitiesbecomes and Statisticsa counting Probabilities and Statistics Computing problem. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Situations in which we can and apply the above theoretical result Probabilities and Statistics Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics must be carefuly distinguished as the elementary random Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics events and areStatistics not always equiprobable. witness and weStatistics can Probabilities Probabilities and Statistics As aProbabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics use the simmetry priciple: similar events must have equal Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics probabilities. Equiprobable elementary random events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Example. we withdraw a ball a box, Probabilities any ball and hasStatistics the ProbabilitiesWhen and Statistics Probabilities and from Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics same probility to be chosen. If the box contains Probabilities balls of different Probabilities and Statistics Probabilities and Statistics and Statistics colors: two white andProbabilities three black, a black Probabilities ball has and a different Probabilities and Statistics and Statistics Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics probability of occurrence than a white ball. Probabilities Probabilities and Statistics Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Example. we toss a Probabilities fair die, and theStatistics probability of occurrence of a Probabilities andIf Statistics Probabilities and Statistics Probabilities and Statistics and Statistics Probabilities number is always the sameProbabilities for all the numbers from 1 2 and Statistics 6 . Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ButProbabilities if threeand faces of the die have and number the remaining Statistics Probabilities Statistics 1, and Probabilities and Statistics Probabilities Statistics Probabilities and Statistics Probabilities and Statistics have 2, 3,andand 4 respectively, then the elementary outcomes have Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics different and probabilities. Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics | f ; ;:::; g | Seminar’s Exercises Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics Random Events:Probabilities I.1., I.2. (a), I.3., I.4.Probabilities (a,Probabilities c), I.6. (a, b, Probabilities and Statistics and Statistics and Statistics d) , I.7 and (b,c) Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probability function: II.7., and II.8., II.13., II.17., II.19., Probabilities and Statistics Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.20., II.22. (b, c) Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Reserve: II.24., II.27., II.30. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics End Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Exercises - Random Events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics I.1. Three şi 3 flip,and inStatistics this order,Probabilities a coin. The winner Probabilities and players, Statistics 1, 2Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics is the player which gets the tail first. The set of random events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics is (0 for head 1 for tail) Probabilities andand Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 1 Probabilities 01 001 0001 Probabilities and Statistics and Statistics 0000Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics a) Determine the random events defined Probabilities like follows: Ai Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics player i wins the game, i 1 3. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics b) Determine the following random events Probabilities A1Probabilities A3and , AStatistics A2 , 1 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics A1 A2and Statistics A3 , and AProbabilities A A . 1 2 3 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics c)Probabilities Show that A A A . True false: A1Probabilities A3 and A2Statistics ? and Statistics and or Statistics 1 3 Probabilities 2 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics I.2.Probabilities A limo and which deserves an airport has a capacity of 6andplaces Statistics Probabilities and Statistics Probabilities Statistics Probabilities and Statistics Probabilities Statistics Probabilities Statistics and transports passangers fromandthe three hotels of theandairport. =f ; ; ; ;:::; = [ [ [ [ ; [ g = [ [ [ = Exercises - Random Events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics andthe Statistics Probabilities andare Statistics Probabilities and Statistics a)Probabilities Describe sample space if we interested in the number Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics of passangers from each of theandhotels. Probabilities and Statistics Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities Describe sample space if we interested only inandthe in andthe Statistics Probabilities and are Statistics Probabilities Statistics Probabilities and Statisticsof passangers Probabilities which and Statistics Probabilities and Statistics the number get to the airport. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities I.3. Two dice are tossed (one red and one black). LetandAStatistics be the Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics random event "the sum is odd”, "at least one die 1, and Probabilities and Statistics Probabilities andB Statistics Probabilities andis Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics C ”the sum is 5”. Describe the following random events using Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics elementary B, A C B C , A Probabilities C , A and B ,Statistics and Probabilities outcomes and Statistics A Probabilities and, Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics A Probabilities B C . and Statistics Probabilities and Statistics Probabilities and Statistics I.4. Let A B be two randomandevents withanda Statistics random Probabilities andand Statistics Probabilities Statisticsassociated Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics experience. Show that Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a) A and B Statistics A B , AProbabilities B A B; Probabilities and Statistics Probabilities and Statistics Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities A B andA B , AProbabilities B B A B ; Probabilities Probabilities and Statistics and Statistics and Statistics = \ \ c) = \ [ \ \ [ = \ \ = [ [ = \ \ = \( [ ) A \ B = B \ (A \ B ), A \ B = B n (A \ B ). \ Exercises - Random Events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics I.5. Let A B be two randomandevents withanda Statistics random Probabilities andand Statistics Probabilities Statisticsassociated Probabilities Probabilities and Statisticsas much Probabilities and Statistics Probabilities and Statistics experience. Reduce as possible the following expressions Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a)Probabilities A B and Statistics A B ; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities A B and Statistics A B Probabilities A B .and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics Probabilities and Statistics I.6.Probabilities Let A, and B ,Statistics and C be Probabilities three random events associated with a Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics random experience. (asandexpression inProbabilities A, B , and andStatistics C) Probabilities and Statistics Determine Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics the following random events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a)Probabilities only A and occurs among three events; Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b) A and C occur but not B ; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics c) at least two of theProbabilities three events occur; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics d) exactly one of theProbabilities three events occurs; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics e) at most three of them occur. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ( [ )\( [ ) ( [ )\( [ )\( [ ) Exercises - Random Events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics Probabilities and Statistics I.7.Probabilities Let A and B randomProbabilities events. and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics and Statistics Probabilities and A A BProbabilities A B and B A B A Statistics B ; a)Probabilities Show that Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities Show that A B A B andA B AProbabilities B . and Statistics and Statistics Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics c)Probabilities Consider Let A beProbabilities the set and of Statistics outand rolling Statistics a six-sided Probabilitiesdie. and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics comes where the roll is an odd number. Let B be the set Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics of outcomes the rolland is Statistics less than 4.Probabilities Calculate the sets Probabilities and StatisticswhereProbabilities and Statistics Probabilities Statistics Probabilities in and part Statistics and Statistics on bothandsides of the equality b), andProbabilities verify that the Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics equalityandholds. Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics = ( \ )[( \ ) = ( \ )[( \ ) \ =( \ )[( \ )[( \ ) Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.1. Weand consider experience 1 and 2 Statistics6 . Probabilities Statistics a random Probabilities and Statisticswith Probabilities and Statistics Probabilities and Statistics Probabilities WeProbabilities know that P 1 2, P 1 2 3 1 2 3 4 and5Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 5 6Probabilities and P and Statistics 1Probabilities 6. Weand define 1 2 Statisticsthe following Probabilities random and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics events A 1 2 4 5 ,B 3 4 5 6 and C 3 4 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Statistics a) Events A and B are compatible? What Probabilities about Aand and C? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b) Compute P A B C , P A B , P A C and P B A . Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics a random Probabilities and StatisticshavingProbabilities and Statistics II.2. Weand consider experience 1 2 6 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics We knowand that P P 1 6 and Probabilities P Probabilities Statistics 1 Probabilities 2and Statistics 2 and 3 Statistics 4 5 and Statistics Probabilities and Statistics and Statistics 2 3.Probabilities We define the following random eventsProbabilities AProbabilities 3 Statistics 4 5 , Probabilities and Statistics Probabilities and Statistics and B Probabilities . Probabilities2 and6Statistics Probabilities and Statistics 1 3and 4Statistics 5 and C Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a)Probabilities Events and A Statistics and B areProbabilities complementary? aboutand BStatistics and and Statistics What Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics C? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b) Calculate P A CProbabilities , P A and B Statistics , P B A Probabilities C andand P Statistics A C . = = f! ; ! ; : : : ; ! g (f! ; ! ; ! g) = = (f! ; ! ; ! ; ! ; ! g) = (f! ; ! g) = = = f! ; ! ; ! ; ! g = f! ; ! ; ! ; ! g = f! ; ! ; !5g. ( [ ( n )) ( ) ( [ ) ( n ) = f! ; ! ; : : : ; ! g. (f! g) = (f! g) = = (f! ; ! ; ! ; ! g) = = = f! ; ! ; ! g = f! ; ! ; ! ; ! g = f! ; ! g ( [ ) ( ) ( n( \ )) ( ) Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.3. Let A B be two random events associated with a Probabilities and and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics such Probabilities random experience that PandBStatistics A 1 Probabilities 3, P A andBStatistics 1 6 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and P A B 1 3. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and P Statistics a)Probabilities Compute A , P AProbabilities B , P and A Statistics B and PProbabilities A B and . Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities Events and A Statistics and B are complementary? B are compatProbabilities and StatisticsA and Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ible? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.4. Let and B be two random associated with a Probabilities and A Statistics Probabilities and StatisticseventsProbabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics random experience such that P A B 1 Probabilities 6, P A andBStatistics 1 6 Probabilities and Statistics Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics andProbabilities P B A 1 3. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics A B and P A andBStatistics . a)Probabilities Compute P A B , P A B , P and Statistics Probabilities and Statistics Probabilities Probabilities and Statistics Probabilities and StatisticsA andProbabilities and Statistics b) Events A and B are compatibile? B A are compleProbabilities and Statistics Probabilities and Statistics Probabilities and Statistics mentary? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics II.5. Let A and B be two random events such that P AandB Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 2 7,Probabilities P A and B Statistics 1 7 andProbabilities P A Band Statistics 5 7. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ( n )= ( n )= = ( ) ( ) ( [ ) = ( n )= = ( \ )= = ( n ) ( n )= = ( [ ) ( ) ( \ ) ( \ )= n = ( n ) ( n )= ( \ )= = ( )= = a) Calculate P (A), P (B ), P (A [ B ), P (A n B ) and P (B n A). = b) Events A and B are complementary? But compatible? Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.6. Let A B be two random events associated with a Probabilities and and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics such Probabilities random experience that PandBStatistics A 2 Probabilities 5, P A andBStatistics 1 5 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and P A 5 8. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and P Statistics a)Probabilities Compute B , P AProbabilities B , P and A Statistics B and PProbabilities A B and . Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities Events and A Statistics and B are Probabilities complementary? and Statistics But compatible? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.7. Let A and B be two random events associated with a Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics random experience that PandAStatistics B 1 Probabilities 6, P A andBStatistics 5 6 Probabilities and Statistics such Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and P B A 1 3. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a)Probabilities Calculate P A , P AProbabilities B , P and AStatistics B and P A B and . Statistics and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities B and A are complementary? A and A B are compatible? and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics a random Probabilities and Statisticssuch that Probabilities and1Statistics II.8. Weand consider experience 2 6 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics We know that P 1 3, P 5 6 2 Probabilities 3 3 4 and 5 Statistics Probabilities and Statistics and Statistics 1 2Probabilities Probabilities Statistics random Probabilities and Statistics şi PProbabilities 1 6. Define the and following events A 4 5and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ,B C 2Probabilities 3 4 and 5 Statistics 4 Probabilities 5 6 and 1 Probabilities 4 5 . and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ( )= = ( n )= = ( ) ( ) ( [ ) ( n )= ( n )= = ( ) ( ) ( \ ) (f! ; ! g) = (f! ; ! g) = f! ; ! ; ! ; ! g = ( \ )= ( n ) = n ( [ )= = f! ; ! ; : : : ; ! g. (f! ; ! ; ! ; ! ; ! g) = = = = f! ; ! ; ! g = = f! ; ! ; ! g ) ( = ( n ) a) A and B are complementary? What about A and B . b) Compute P A [ C , P C n B and P A B . ( = ) ( ) Exercises - Probability Function = f! ; ! ; : : : ; ! g (f! ; ! g) = = Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.9. We consider a random experience with 1 2 and Statistics 6 . Probabilities and Statistics Probabilities and Statistics Probabilities Probabilities Statistics and 1 Statistics We knowandthat P 7 12, P Probabilities 3 and 2 Probabilities 4 6 and Statistics 3 5 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics P 5 12. We define the following random events 2 and 4 Statistics Probabilities Probabilities and Statistics Probabilities and Statistics A Probabilities ,B and C Probabilities 1 3and 5Statistics 1Probabilities 2 4 and6 Statistics 1 6 .and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a)Probabilities Events and A Statistics şi C are complementary to each other? Events A Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and B sunt compatible? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b) Calculate P A Probabilities B C and , PStatistics B C , PProbabilities C A ,andPStatistics B A Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics B C . Probabilities and Statistics and and P Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.10. Consider experience with only six possible Probabilities and Statisticsa random Probabilities and Statistics Probabilities and StatisticsreProbabilities and Statistics Probabilities and Statistics Probabilities sults s1 s2 s6 . We know that P s2 s5 1and6Statistics and Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 2P Probabilities s1 s4 and Statistics P s2 s3 Probabilities s5 s6 ; and define the following random Statistics Probabilities and Statistics Probabilities ands2Statistics events Aand Statistics s1 s2 s4 sProbabilities s3 s5 s6 Probabilities and C and Statistics s1 s4 . 5 , B Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Statistics a) Events A and B are compatible? What Probabilities about Aand and C? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Statistics and b) Compute P B Probabilities A C ,andPStatistics C B , PProbabilities A B and (f! ; ! ; ! g) = = (f! ; ! g) = = = f! ; ! ; ! g = f! ; ! ; ! ; ! g = f! ; ! g ( [ ( \ )) ( ) ( n ) ( n ) ( \ ) (f = f ; ;:::; g ; g) = (f ; ; ; g) =f ; ; ; g =f ( ) P A B . (f ; ; ; g ; g) = = =f ( [ ( \ )) ( n ) ( \ ) ; g Exercises - Probability Function =f g Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.11. Consider a randomProbabilities experience having s1 s2 and Statistics s6 . Probabilities and Statistics and Statistics Probabilities Probabilities Statistics and 2Statistics We knowandthat P s1 Probabilities s3 s6 and Statistics 2 5, P sProbabilities 5 and 2 s4 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics P s2 s4ands6Statistics 3 5. Probabilities We define the following random events Probabilities and Statistics Probabilities and Statistics and Statistics and Statistics A Probabilities s5 s6 and , BStatisticss2 s4 Probabilities s5 s6 and C s1 s3Probabilities s5 . Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statisticsto each Probabilities Statistics a)Probabilities Events and A Statistics and C are Probabilities complementary other? and Events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics A and B compatible? Probabilities and are Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities Calculate P B C ,Probabilities P A Cand and P C Probabilities A. and Statistics Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and StatisticsreII.12. Consider a random experience with only six possible Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics sults s1 s2 s6 . Weandknow s1 s2 and s4Statistics s5 Probabilities and Statistics Probabilities Statisticsthat P Probabilities Statistics and1Statistics Probabilities and Statistics 2P Probabilities s3 s6 andand P s3 sProbabilities s 2. Let us define the fol4 6 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics lowing random events AProbabilities s1 and s2 Statistics s5 , B s4 and Probabilities and Statistics Probabilities andC Statistics Probabilities Probabilities and Statistics Probabilities and Statistics s1 Probabilities s2 s3ands5Statistics s . and 6Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Statistics a) Events A and B are compatible? What Probabilities about Aand and C? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities b) Compute P B C , P A andBStatistics , P C AProbabilities and and Statistics (f ; =f (f ; ; (f g) = = ; ; g) = = g =f ; ; ; g =f ; g) =f ; g) g ; ; ( [ ) ( ) ( n ) g (f ; ;:::; f ; ; ; ; g (f (f g) = = =f ; ; g ; ; ; ;:::; = = ; ; ; =f g ( [ ) ( ) ( n ) P (B \ (A [ C )). g) = = Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.13. Let us a random sample Probabilities andconsider Statistics Probabilitiesexperience and Statistics whose Probabilities andspace Statistics Probabilities and and Statistics Probabilities and Statistics is of the following functions can 1 Statistics 2 3 4 Probabilities 5 . What Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics be a probability on ? Why? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 1 1 1 1 Probabilities and1Statistics Probabilities and Statistics Probabilities and Statistics a) P and P and3 Statistics P 4Probabilities Pand 5Statistics ; 1 StatisticsP 2 Probabilities Probabilities 3 9 3 9 9 Probabilities and 1 Statistics Probabilities and Statistics Probabilities 1 1 1 and Statistics Probabilities Statistics Probabilities and Statistics Probabilities andP Statistics b) P and P P P 1 2 3 4 5 8 4 2 and Statistics Probabilities and 4 Statistics Probabilities and Statistics Probabilities 3 and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics ; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 8 Probabilities and Statistics Probabilities and Statistics Probabilities 1 1 1 1 and Statistics3 Probabilities c)Probabilities P 1 and Statistics P 2 P 3 and Statistics P 4 Probabilities P 5and Statistics . 2 8 8 8 and Statistics8 Probabilities and Statistics Probabilities and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics II.14. Let and A Statistics and B be Probabilities two random events associated with a Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics random experience such Probabilities that P A B 1 4, P A 1 3, Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and P B 1 2. Compute P A B , P A B , P B A , Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics andP A B . Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.15. Let and A Statistics and B be Probabilities two random events associated with a Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics = f! ; ! ; ! ; ! ; ! g (! ) = ; (! ) = (! ) = ; (! ) = (! ) = ( )= ( [ ) ; = (! ) = ; ; ; (! ) = ; (! ) = ; (! ) = (! ) = ; (! ) = ; (! ) = (! ) = ; (! ) = ; (! ) = ( \ )= = ( )= = ( [ ) ( n ) ( n ) ( )= ( ) = 1=6 ( n A), random experience such that P A [ B 3=4, P A \ B şi P B 4=5. Compute the probabilities for A, A, B ( )= Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics Probabilities Statistics events Probabilities and A Statistics II.16. Let Aandand B be two disjointandrandom with P Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 0 3 Probabilities and P Band Statistics 0 5. What is the probability of Probabilities the eventand Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a)Probabilities exactly and one of the two events and occur? Statistics Probabilities Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities B occurandbut A doesn’t ? Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics c)Probabilities both events occur? Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.17. Two and senior, s1 and Probabilities s2 and three juniors j1Probabilities , j2 and jand in 3 are Probabilities Statistics and Statistics Statistics Probabilities and Statistics Probabilities andisStatistics Probabilities and Statistics a chess tournament. Any senior twice as likely to win as any Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics junior. and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a) Findandthe chances that a junior wins the Probabilities tournament. Probabilities Statistics Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b) What the probability that s1 or j1 wins the tournament? Probabilities andis Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics : ( )= : ( )= Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Probabilities Statistics and Statistics II.18. Outand ofStatistics the students in a and class, 60 areProbabilities geniuses, 70 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics loveProbabilities chocolate, and 40 fall into both categories. Determine the and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics probability that a randomly selected student is neither a genius Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics nor a chocolate lover. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics signs Probabilities Statistics Probabilities II.19. US and president in to and law a resolution onlyandifStatistics this Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics resolution Congress and Senate. 60 andof all Probabilitiesisandvoted Statisticsin both Probabilities and Statistics Probabilities Statistics Probabilities andpass Statistics Probabilities Probabilities Statistics resolution the Congress, 70and Statistics pass the Senate, andand80 pass Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics at least of one of the Probabilities two chambers. the probability Probabilities and Statistics and StatisticsCompute Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics that Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a) a certain resolution comes and to Statistics president, Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities a certain resolution passes exactly one ofProbabilities the twoand chambers. Probabilities and Statistics Probabilities and Statistics Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics % % % % % % Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.20. A student has toProbabilities choose and two between Probabilities three faculative Probabilities and Statistics Statistics and Statistics Probabilities Statistics Maths, Probabilities and StatisticsHe chooses Probabilities and Statistics courses: and French, and History. History with Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics probability 5 8, French with probability 5 8, and chooses both Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Statistics Probabilities and1Statistics and Statistics History and and French with probability 4. What isProbabilities the probability Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics thatProbabilities he willandchoose Butandthe probability that and heStatistics will Statistics Maths? Probabilities Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics choose History or French? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.21. Three horses compete in Statistics arace: H1Probabilities , H2 , and 3 . H1 Probabilities and Statistics Probabilities and and H Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and has twice the chances to win compared to H2 , and H2 twiceStatistics the Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics chances to win compared to H . 3 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a)Probabilities What are the probabilities of winning of the three horses? and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and is Statistics Probabilitiesthat and Statistics Probabilities and Statistics b) What the probability the first or the second horse Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics doesn’t win? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.22. Let A and B beProbabilities two random events. Show that Probabilities and Statistics and Statistics Probabilities and Statistics and Statistics and Statistics a)Probabilities P A B 1 P AProbabilities P Band Statistics P A B Probabilities ; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics = = = ( \ )= ( ) ( )+ ( \ ) b) P (A [ B ) = P (A n B ) + P (A \ B ) + P (B n A); c) P (AB ) = P (A) 2P (A \ B ) + P (B ). Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.23. Prove the following identities (A, B , and C are Probabilities and Statistics Probabilities and Statistics Probabilities and three Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics random events) Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics B 1; and Statistics a)Probabilities P A B P A BProbabilities B and AStatistics P A Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities P A B P A P A B 1; and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Probabilities c) P Aand Statistics B C P A PandAStatistics B P Probabilities A B and CStatistics 1; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and A B PProbabilities A P Band. Statistics d) P Aand Statistics B P A Probabilities P B P Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and A Statistics Probabilities and Statistics Probabilities and Statistics II.24. Let 1 , A2 şi A3 be three random events. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and the Statistics Probabilities andholds Statistics Probabilities and Statistics a)Probabilities When following relation and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities P A1 and A2Statistics A3 PProbabilities A1 A2and Statistics P A1 A3 Probabilities P A1 and A2Statistics A3 Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b) Show if A Probabilities B C and Statistics , then Probabilities andthat, Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ( ( ( ( \ \ \ \ ) + [( n ) [ ( n )] + ( \ ) = )+ ( )+ ( \ )= \ )+ ( )+ ( \ )+ ( \ \ )= ) ( ) ( )= ( \ ) ( ) ( ) ( \( [ )) = ( \ )+ ( \ )+ ( \ \ )? \ \ =? P [(A[B )\(B [C )\(C [A)] = P (A\B )+P (B \C )+P (C \A): Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.25*. Two students X Probabilities and Y follow a courseProbabilities for which one Probabilities and Statistics and Statistics and Statistics Probabilities Statistics Probabilities and Statistics and Statistics could getandone of the three grades: A, B or Probabilities C. The probability Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics that X gets B is 0 3; the probability that Y Probabilities gets B isand0Statistics 4. The Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics probability that neither get A but at least one gets B 0 1. What Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics is the probability that at Probabilities least oneand gets B but none of them gets Probabilities and Statistics Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics C? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.26*. In a box thereProbabilities are ten and balls numbered from 1andtoStatistics 10; we Probabilities and Statistics Statistics Probabilities Probabilities and Statistics Statistics and Statistics randomly select two balls Probabilities from theand box. What isProbabilities the probability Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics thatProbabilities the sum the numbers on the if andof Statistics Probabilities and balls Statisticsis oddProbabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a)Probabilities the twoand balls are simultaneously selected? Probabilities and Statistics Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities the twoand balls are selected one and byStatistics one without removing? Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics c)Probabilities the twoand balls are selected one and byStatistics one and removed? Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics : : : Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.27*. For a biased die Probabilities the probability of occurrence of a face Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities andon Statistics Statistics is proportional with the number that face.Probabilities We tossandthe die. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities andis Statistics Probabilities and Statistics Probabilities and Statistics a) What the probability that we get an even number? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities andis Statistics Probabilities and Statistics Probabilities and Statistics b) What the probability that we get a prime number? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Probabilities Statistics Probabilities andthe Statistics II.28. Aand dieStatistics is weighted so thatand the even numbers have same Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics chance ofandappearing, the odd numbers the sameandchance Probabilities Statistics Probabilities and Statisticshave Probabilities Statistics of Probabilitiesand and Statistics Statisticsas likely Probabilities and Statistics appearing, each even Probabilities number and is twice to appear as Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics anyProbabilities odd number. Find the probability that and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a)Probabilities a primeandnumber Statistics appears; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities an evenand number Statistics appears; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics c)Probabilities an odd and prime numberProbabilities appears. Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.29. A six-sided die is Probabilities loaded inanda Statistics way that each evenandface is Probabilities and Statistics Probabilities Statistics Probabilities Statistics Statistics Probabilities and Statistics twice as and likely as each Probabilities odd face.andAll even faces are equally likely, Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics as are all odd faces. Find the probability that the outcome is Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Statistics Probabilities and Statistics Probabilities and Statistics a)Probabilities an odd and number; Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities less than and4; Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics c)Probabilities an evenand prime number. Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and times; Statistics which Probabilities andlikely: Statistics a II.30. A fair die is rolled three is more Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics sum of 11 a sum of Probabilities 12? (de and Méré to his friend Pascal.) Probabilities andor Statistics Statistics Probabilities and Statistics Probabilities Probabilities Probabilities II.31*. If PandAStatistics0 9 andP B and0Statistics 8, show that P A andBStatistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 0 7.Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a)Probabilities Prove that P A B Probabilities P A and Statistics P B 1. Probabilities and Statistics and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b) Prove by induction the Bonferroni inequality: given n ranProbabilities and Statistics Probabilities and Statistics Probabilities and Statistics dom events A A A we have 2Probabilities n and Statistics Probabilities and Statistics 1 Probabilities and Statistics ! Probabilities and Statistics \ and Statistics Probabilities and Statistics n Probabilities n X Probabilities and Statistics and Statistics Probabilities and Statistics Ai P Ai n 1 P Probabilities i 1 i 1 : ( )= ( )= : ( \ )> : ( \ )> ( )+ ( ) ; ;::: = > = ( ) + : Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.32*. Let and usStatistics consider a Probabilities random experience an infinitely Probabilities and Statistics having Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics countable elementary results: Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 1 2 n Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statistics Probabilities Statistics a)Probabilities Show that is not possible thatand allStatistics elementaryProbabilities randomandevents Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics have the positive probability p. Probabilities andsame Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics b)Probabilities True orandfalse: random have positive Statisticsall elementary Probabilities and Statistics events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics probabilities? Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a random Probabilities and Statistics having Probabilities and Statistics II.33*. Consider experience the sample space Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics . Prove that d 0 1 , defined by d A B Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics P A B , isanda Statistics metric on the family . Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics = f! ; ! ; : : : ; ! ( ) ; : : :g: : P ( ) P ( ) ! [ ; ] P ( ) ( ; )= Exercises - Probability Function Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics II.34**.andLet A1 A2 Probabilities An be having the folProbabilities Statistics and random Statistics events Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics lowing properties: Probabilities and[ Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics 1 n; (i) A A , i j i and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics j and i Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics (ii) Ai and AStatistics A , 1 i j k n. j k Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Prove that ! Probabilitiesn and Statistics Probabilities and Statistics Probabilities and Statistics n [ X Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics a)Probabilities 2P AStatistics P Probabilities Ai ; i and and Statistics Probabilities and Statistics Probabilities and and Statistics Probabilities and Statistics i Statistics 1 i Probabilities 1 and Statistics Probabilities and Statistics and Statistics b)Probabilities for every 1 i n, there exists at most Probabilities oneProbabilities j iandsuch that Probabilities and Statistics Probabilities and Statistics Statistics Ai Ajand Statistics . Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics ; \ 6= = \ ;:::; 8 = ; \ =? 8 6 = = 6 6 =6 ? < < 6 ( ) 6= Bibliography Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Bertsekas, D. P., J.Probabilities N. Tsitsiklis, Introduction toand Statistics Probabilities and Statistics and Statistics Probabilities Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probability, Athena Scietific, 2002. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Gordon, H., Discrete Probability, Springer Verlag, New Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics York, 1997. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics and Statisticsof Probability, Probabilities and Statistics Lipschutz, S., TheoryProbabilities and Problems Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Schaum’s Outline Series, McGraw-Hill, 1965. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Ross, S. M., A First Course in Probability , Prentice Hall, Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics1998. Probabilities and Statistics Probabilities and Statistics 5th edition, Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and C. Statistics Probabilities Statistics Probabilities and Statistics Stone, J., A Course in and Probability and Statistics, Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Duxbury Press, 1996. Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics Probabilities and Statistics
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