1. Chapter 4 Probability Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

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

3. 4.1 Probability and Sample Spaces 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 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 LO4-1: Define a probability and a sample space. 4-3

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 LO4-1 4-4

5. Assigning Probabilities to Sample Space Outcomes 1. Classical method ◦ For equally likely outcomes 2. Relative frequency method ◦ Using the long run relative frequency 3. Subjective method ◦ Assessment based on experience, expertise or intuition LO4-1 4-5

6. 4.2 Probability and Events 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 LO4-2: List the outcomes in a sample space and use the list to compute probabilities. 4-6

7. 4.3 Some Elementary Probability Rules 1. Complement 2. Union 3. Intersection 4. Addition 5. Conditional probability 6. Multiplication LO4-3: Use elementary profitability rules to compute probabilities. 4-7

8. 4.4 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 LO4-4: Compute conditional probabilities and assess independence. 4-8

9. 4.5 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 LO4-5: Use Bayes’ Theorem to update prior probabilities to posterior probabilities (Optional). 4-9

10. Bayes’ Theorem Continued LO4-5 4-10

11. 4.6 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)! LO4-6: Use elementary counting rules to compute probabilities (Optional). 4-11