Aim #94 ­ How do determine the probability of an event occurring? Module 4: Probability & Stats
Do Now: Determine the probability of spinning
1. a 3
2. an odd number 3. a 7
• The probability of an event E occurring is given by the ratio: P(E) = n (E) , where: n(E) is the number of outcomes that fall into the event E
n (S) n(S) is the number of outcomes that fall in the sample space.
• A probability must always lie between ______ and ______, and ______ means the event is certain and _______ means the event is impossible. • Theoretical Probability of an event is the number of ways the event can occur (favorable outcomes) divided by the number of total outcomes.
Example: A fair coin is flipped three times and the result is noted each time. (a) List all of the outcomes as ordered triples. How many of them are there?
(b) Find each of the following probabilities (1) P(all heads)
(2) P(exactly 2 heads)
Ex 2) The accompanying figure is a square. The interior sections are formed using congruent squares. If this figure is used as a dart board, what is the probability that the dart will hit the shaded blue region? (3) P(all heads or all tails)
What is Empirical Probability
? The Empirical (or Experimental) Probability of an event is an... "estimate" that an event will occur based upon how often the event occurred after collecting data from an experiment.
6) A survey was done to determine which of three sodas was preferred by people in a taste test. Results are shown below.
a) Find the empirical probability that a person selected at random from this group would prefer soda B. Express your answer as a fraction & also to nearest hundredth.
b) Find the empirical probability that a person selected at random from this group would NOT prefer soda A.
Sets & Probability • A set is simply a collection of things (numbers, objects, etc.) that satisfy a well­defined criteria. The things that are contained in the set are called the elements of the set.
• A subset is any set whose elements are all contained within another set.
• The complement of set A will be all of the events in the sample space S that do not fall into set A. The complement of set A is denoted as Ac or A'.
Example 7) Set A is defined as all integers greater than zero and less than 10. Set A: ____________________________________
Subset B is all evens in set A
Set B' is the complement of set B.
B: ________________
C: __________________
A Venn diagram is a diagram that uses circles to illustrate the relationships among sets. Example 8) The data from a survey of 50 students is shown in the Venn diagram. The students were asked whether or not they were taking a foreign language and whether or not they played a sport. (a) How many students are taking a foreign language?
(b) How many students play a sport?
(c) How many students do both?
(d) How many students do not play a sport and do not take a foreign language?
(e) If a student is randomly chosen from this set, what is the probability that this student plays a sport, but does not take a foreign language?
9) Consider rolling a single six­sided die and recording the result. Let set A be the event of rolling a number greater than 4 and let set B be the complement of set A.
(a) Draw a Venn Diagram that illustrates the sample space, S, and sets A and B.
(b) Find P(A) (c) Find P(B)
(d) What is the sum of P(A) and P(B)? Why does this make sense?
Sum it Up
10) If the probability a red marble is drawn from a bag is 3/17, find the probability a red marble will not be drawn. 11) If the probability that it will rain tomorrow is 20%, find the probability that it won’t rain.
1) A bag contains 5 red marbles and 6 black marbles. One marble is drawn from the bag.
a. What is the probability that the ball is red?
b. What is the probability that the ball is black?
2) A deck of cards contains 52 cards, 13 of each suit and 4 of each type. One card is drawn. What is the probability that the card is
a. an Ace?
b. black?
c. a diamond?
d. a red King?
"OR"
P(A or B) = P(A) + P(B) ­ P(A and B)
3) Using the same deck of cards, determine
a. P(7 or 10)
b. P(red or queen)
c. P(black king or a club)
Probability of a Compound Event
more than one event
"AND"
P(A and B) = P(A) P(B)
4) In a toss of three coins, what is the probability of obtaining a head, a tail and then a head?
We can draw a tree diagram
= We can also list the sample space
5) In a toss of three coins, what is the probability of obtaining exactly two tails?
Recall the sample space