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Chapter 9 Probability Copyright © 2013, 2010, and 2007, Pearson Education, Inc. 9-4 Odds, Conditional Probability, and Expected Value Computing Odds Conditional Probabilities Expected Value Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Computing Odds Let P(A) be the probability that A occurs and P(A) be the probability that A does not occur. Then the odds in favor of an event A are and the odds against an event A are Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Computing Odds In the case of equally like outcomes, odds in favor odds against Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 9-12 Find the odds in favor of the event occurring: a. Rolling a number less than 5 on a die 4 : 2 or 2 : 1 b. Tossing heads on a fair coin 1:1 Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 9-12 cont Find the odds in favor of the event occurring: c. Drawing an ace from an ordinary 52-card deck 4 : 48 or 1 : 12 d. Drawing a heart from an ordinary 52-card deck 13 : 39 or 1 : 3 Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 9-13 Find the probability of making totally black copies if the odds are 3 to 1 against making totally black copies. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Computing Odds If the odds in favor of event E are m : n, then If the odds against event E are m : n, then Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Conditional Probabilities If A and B are events in sample space S and P(A) 0, then the conditional probability that an event B occurs given that event A has occurred is given by Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 9-14 What is the probability of rolling a 6 on a fair die if you know that the roll is an even number? If event B is rolling a 6 and event A is rolling an even number, then Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Expected Value If, in an experiment, the possible outcomes are numbers occurring with probabilities respectively, then the expected value (mathematical expectation) E is given by Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Expected Value Expected value can be used to predict the average result of an experiment when it is repeated many times. Expected value cannot be used to determine the outcome of any single experiment. Fair game: When payoffs are involved and the net winnings are $0 (the expected value minus cost to play a game of chance), the game is a fair game. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 9-15 Suppose you pay $5.00 to play the following game. Two coins are tossed. You receive $10 if two heads occur, $5 if exactly one head occurs, and nothing if no heads appear. Is this a fair game? That is, are the net winnings $0? The net winnings are $0, so this is a fair game. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.