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Chapter 4 Probability McGraw-Hill/Irwin Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Probability 4.1 4.2 4.3 4.4 4.5 4.6 The Concept of Probability Sample Spaces and Events Some Elementary Probability Rules Conditional Probability and Independence Bayes’ Theorem (Optional) Counting Rules (Optional) 4-2 LO 1: Explain what a probability is. 4.1 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 4-3 LO1 Probability If E is an experimental outcome, then P(E) denotes the probability that E will occur and: Conditions 0 P(E) 1 such that: 1. 2. If E can never occur, then P(E) = 0 If E is certain to occur, then P(E) = 1 The probabilities of all the experimental outcomes must sum to 1 4-4 LO1 Assigning Probabilities to Experimental Outcomes Classical Method Long-run relative frequency For equally likely outcomes In the long run Subjective Assessment based on experience, expertise or intuition 4-5 LO 2: List the outcomes in a sample space and use the list to compute probabilities. 4.2 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 4-6 LO 3: Use elementary profitability rules to compute probabilities. 1. 2. 3. 4. 5. 6. 4.3 Some Elementary Probability Rules Complement Union Intersection Addition Conditional probability Multiplication 4-7 LO3 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 4-8 LO3 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(AB) = P(A) + P(B) If A and B are not mutually exclusive: P(AB) = P(A) + P(B) – P(A∩B) where P(A∩B) is the joint probability of A and B both occurring together 4-9 LO 4: Compute conditional probabilities and assess independence. The probability of an event A, given that the event B has occurred, is called the conditional probability of A given B 4.4 Conditional Probability and Independence Denoted as P(A|B) Further, P(A|B) = P(A∩B) / P(B) P(B) ≠ 0 4-10 LO4 The General Multiplication Rule There are two ways to calculate P(A∩B) Given any two events A and B 1. 2. P(A∩B) = P(A) P(B|A) and P(A∩B) = P(B) P(A|B) 4-11 LO4 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) 4-12 LO 5: Use Bayes’ Theorem to update prior probabilities to posterior probabilities (optional). 4.5 Bayes’ Theorem S1, S2, …, Sk represents k mutually exclusive possible states of nature, one of which must be true P(S1), P(S2), …, P(Sk) 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 Si, given the experimental outcome E, is calculated using the formula on the next slide 4-13 LO5 Bayes’ Theorem Continued P(Si E) P(Si|E) = P(E) P(Si )P(E|S i ) P(E) P(Si )P(E|S i ) P(S1 )P(E|S1 )+P(S 2 )P(E|S 2 )+ ...+P(Sk )P(E|S k ) 4-14 LO 6: Use elementary counting rules to compute probabilities (optional). 4.6 Counting Rules (Optional) A counting rule for multiple-step experiments (n1)(n2)…(nk) A counting rule for combinations N!/n!(N-n)! 4-15