1. © 2010 Pearson Education, Inc. All rights reserved Chapter 9 9 Probability

2. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 2 NCTM Standard: Data Analysis and Probability  K–2: Children should discuss events related to their experience as likely or unlikely. (p. 400)  3–5: Children should be able to “describe events as likely or unlikely and discuss the degree of likelihood using words such as certain, equally likely, and impossible.” They should be able to “predict the probability of outcomes of simple experiments and test the predictions.” They should “understand that the measure of the likelihood of an event can be represented by a number from 0 to 1.” (p. 400)

3. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 3 NCTM Standard: Data Analysis and Probability 6–8: Children should “understand and use appropriate terminology to describe complementary and mutually exclusive events.” They should be able “to make and test conjectures about the results of experiments and simulations.” They should be able to “compute probabilities of compound events using methods such as organized lists, tree diagrams, and area models.” (p. 401)

4. Slide 9.4- 4 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 9-4Odds, Conditional Probability, and Expected Value  Computing Odds  Conditional Probabilities  Expected Value

5. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 5 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

6. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 6 In the case of equally like outcomes, odds in favor odds against Computing Odds

7. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 7 Find the odds in favor of the event occurring: Example 9-13 a.Rolling a number less than 5 on a die 4 : 2 or 2 : 1 b. Tossing heads on a fair coin 1 : 1 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

8. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 8 Find the probability of making totally black copies if the odds are 3 to 1 against making totally black copies. Example 9-14

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 9 Computing Odds If the odds in favor of event E are m : n, then If the odds against event E are m : n, then

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 10 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

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 11 Example 9-15 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

12. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 12 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

13. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 13 Expected Value 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.  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.

14. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 14 Example 9-16 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.