Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter 4 Probability

4-2 Probability 4.1 The Concept of Probability 4.2 Sample Spaces and Events 4.3 Some Elementary Probability Rules 4.4 Conditional Probability and Independence 4.5Bayes’ Theorem (Optional) 4.6Counting Rules (Optional)

The Concept of Probability An experiment is any process of observation with an uncertain outcome The possible outcomes for an experiment are called the experimental outcomes Probability is a measure of the chance that an experimental outcome will occur when an experiment is carried out LO 1: Explain what a probability is.

4-4 Probability If E is an experimental outcome, then P(E) denotes the probability that E will occur and: Conditions 1. 0  P(E)  1 such that: If E can never occur, then P(E) = 0 If E is certain to occur, then P(E) = 1 2. The probabilities of all the experimental outcomes must sum to 1 LO1

4-5 Assigning Probabilities to Experimental Outcomes Classical Method For equally likely outcomes Long-run relative frequency In the long run Subjective Assessment based on experience, expertise or intuition LO1

Sample Spaces and Events The sample space of an experiment is the set of all possible experimental outcomes The experimental outcomes in the sample space are called sample space outcomes An event is a set of sample space outcomes The probability of an event is the sum of the probabilities of the sample space outcomes If all outcomes equally likely, the probability of an event is just the ratio of the number of outcomes that correspond to the event divided by the total number of outcomes LO 2: List the outcomes in a sample space and use the list to compute probabilities.

Some Elementary Probability Rules 1. Complement 2. Union 3. Intersection 4. Addition 5. Conditional probability 6. Multiplication LO 3: Use elementary profitability rules to compute probabilities.

4-8 Union and Intersection The union of A and B are elementary events that belong to either A or B or both Written as A  B The intersection of A and B are elementary events that belong to both A and B Written as A ∩ B LO3

4-9 The Addition Rule If A and B are mutually exclusive, then the probability that A or B (the union of A and B) will occur is P(A  B) = P(A) + P(B) If A and B are not mutually exclusive: P(A  B) = P(A) + P(B) – P(A∩B) where P(A∩B) is the joint probability of A and B both occurring together LO3

Conditional Probability and Independence The probability of an event A, given that the event B has occurred, is called the conditional probability of A given B Denoted as P(A|B) Further, P(A|B) = P(A∩B) / P(B) P(B) ≠ 0 LO 4: Compute conditional probabilities and assess independence.

4-11 The General Multiplication Rule There are two ways to calculate P(A∩B) Given any two events A and B 1. P(A∩B) = P(A) P(B|A) and 2. P(A∩B) = P(B) P(A|B) LO4

4-12 The Multiplication Rule The joint probability that A and B (the intersection of A and B) will occur is P(A∩B) = P(A) P(B|A) = P(B) P(A|B) If A and B are independent, then the probability that A and B will occur is: P(A∩B) = P(A) P(B) = P(B) P(A) LO4

Bayes’ Theorem S 1, S 2, …, S k represents k mutually exclusive possible states of nature, one of which must be true P(S 1 ), P(S 2 ), …, P(S k ) represents the prior probabilities of the k possible states of nature If E is a particular outcome of an experiment designed to determine which is the true state of nature, then the posterior (or revised) probability of a state S i, given the experimental outcome E, is calculated using the formula on the next slide LO 5: Use Bayes’ Theorem to update prior probabilities to posterior probabilities (optional).

4-14 Bayes’ Theorem Continued LO5

Counting Rules (Optional) A counting rule for multiple-step experiments (n 1 )(n 2 )…(n k ) A counting rule for combinations N!/n!(N-n)! LO 6: Use elementary counting rules to compute probabilities (optional).