CHAPTER 3: PROBABILITY
Definitions:
Experiment – any trial that produces outcome such as tossing a coin which will result in getting a Head (H)
or Tail (T). In an experiment of sitting for an exam, the outcome is either Pass or Fail.
Outcome – the result of an experiment
Event – a set of outcome or outcomes of an experiment. For example in a game of football, an event can
be defined as winning the game, losing the game, draw or not losing which comprises winning or draw.
Sample space – a set that lists all possible outcomes of an experiment. In an experiment of tossing a coin
the sample space is S = {H, T}
Equally likely outcome – each outcome for an experiment has an equal probability of occurrence.
Finding Probability
There are three conceptual approaches to finding probability. These are Classical Probability, Empirical
Probability and Subjective Probability.
(a) Classical Probability
For many simple experiments, the various outcomes of the experiment have the same probability
occurrence. The classical probability rule applies to finding probabilities from such experiments.
Definition of Classical Probability
Suppose S is a sample space and each outcome in S is equally likely to occur. If A is an event (a subset of
S) then the probability of A is:
P (A) =
=
Example: Consider an experiment in rolling a dice. There are six possible outcomes.
Sample space = {1, 2, 3, 4, 5, 6}, therefore n (S) = 6
Suppose A is the event of getting an odd number
A = {1, 3, 5}, therefore n(A) = 3
Hence the probability of getting an odd number is P (A) = = 0.5
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Tree Diagram – a tree diagram is used to display the outcomes of an experiment which consists of a series
of activities. This ensures us that all logical possibilities are considered. For example, if a coin is tossed
twice, the possible outcomes are {HH, HT, TH, TT}. Using Tree diagram, the possible outcomes can be
displayed in a clearer way.
Example: Three companies, A, B, and C are competing for a contract to build a condominium. The
probabilities that companies A, B, and C will win the contract are 0.25, 0.45 and 0.3 respectively. If
company A, B, and C win the contract, the probability that they will make profits are 0.8,0.9, 0.7
respectively.
(a) Construct a tree diagram based on information given
(b) What is the probability that the companies will make the profit?
(c) If the contract is found to be profitable, find the probability that the contract was given to company A.
Solution:
(b) The probability that the profit will made by the companies is
P (U)
= P (A ∩ U) or P (B ∩ U) or P (C ∩ U)
= 0.2 + 0.405 + 0.21
= 0.815
(c) If the contract is found to be profitable, the probability that the contract will be given to company A is
P (U/A) =
=
= 0.245
(b) Empirical or Relative Probability
Certain experiment or procedures do not have equally likely outcomes. We rely on past data to generate
the data by performing the experiment a large number of times. The relative frequency of the event
happening is then used as the probability of the event.
Definition of Empirical Probability
If an experiment / procedure is repeated n times and an event A is observed k times, then:
P (A) =
=
2
Example: Among 400 randomly selected drivers in the 20 – 24 age brackets, 136 were involved in an
accident the previous year. If a driver in that age bracket is randomly selected, what is the probability
he/she will be involved in an accident? Do you think the value is high enough to be concern?
(c) Subjective Probability
Certain events are not simple events, nor can they be generated. Examples are the probability of a
Tsunami happening, fuel price increase again within the next six months. For these types of events,
subjective judgment based on experience and frequently expert’s advices are used to estimate the
probability.
Definition of Subjective Probability
Subjective probability is the probability assigned to an event based on subjective judgment, experience,
information and belief.
Basic Probability Rules:
1.
0 ≤ P (A) ≤ 1
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For n mutually exclusive events,
3.
If A’ is the complement of A, then P(A’) = 1 – P(A).
Counting Rules – Combination and Permutation
Combination: If there are N elements in a population and we want r a sample size r then the number of
ways of selecting r elements from a total of N elements is given by: nCr
Example: A box contains 20 balls point pen. In how many ways can five pens be selected from the box?
The number of ways of selecting 5 out of 20 ball point pen is 20C5 = 15,504
Permutation: If there are N elements and we want to select r elements but the order in which the elements
are selected is important, then the number of ways this can be done is nPr.
Example: Suppose you have been hired by ASTRO, and part of your job is to determine the shows to be
shown on Monday night. There are 27 shows available and you must select 4 of them. How many different
sequences of four shows are possible?
You need to select r = 4 shows from n = 27 shows. The number of different arrangements possible is
27P4
= 421,200
** When order is important, for example the outcome ab is different from ba, and then we consider them to
be two different events. This is permutation rule. When order is not important then the outcome ab and ba
is considered as one event. This is combination rule.
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Additive rule of Probability
A Venn diagram is occasionally helpful in determining the probability of simple events.
For any two events A and B, the probability of either event A or B happening is denoted P (A ∪ B), where;
P (A ∪ B) = P (A) + P (B) – P (A ∩ B)
P (A ∩ B) represents the probability of both events A and B happening together. If A and B are mutually
exclusive events then P (A ∩ B) = 0
Example: In a group of 250 people, 140 are female, 60 are vegetarian and 40 are female and vegetarian.
What is the probability a randomly selected person from this group is a male or vegetarian?
Mutually Exclusive Events
Events that cannot occur together are called mutually exclusive events. In other words mutually exclusive
events means there are no intersection between those events, that is P (A ∩ B) = Ø (empty set)
Conditional Probability
Conditional probability refers to the probability of an event occurring given the knowledge that some other
events has occurred. The probability that events A occurs given that event B had occurred is
P (A | B) =
Multiplication Rule of Probability
P (A ∩ B) = P (A | B) P (B) or equivalently P (A ∩ B) = P (B | A) P(A)
Example: Suppose student is given eight statements in a quiz. Three of the statements are right while five
are wrong. The student is asked to select two correct statements. What is the probability of getting:
(a) Both correct answers if the student is just guessing?
(b) At least one correct answer?
Solution:
(a) Probability of getting both answers right
P (R1 ∩ R2) = P (R1) P (R2|R1) =
=
(b) Using Tree Diagram
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Independent Events
Two events are said to be independent if the occurrence of one does not affect the probability of
occurrence of the other. If A and B are independent events, then
P (A | B) = P (A) or P (B | A) = P (B) and the multiplicative rule becomes P (A ∩ B) = P (A) P (B)
Bayes Theorem
If events A1, A2, …, Ak are mutually exclusive and for any event B where P(B) ≠ 0
P (Ak | B) =
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