1. Chapter 12 Section 1 - Slide 1 Copyright © 2009 Pearson Education, Inc. AND

2. Copyright © 2009 Pearson Education, Inc. Chapter 12 Section 1 - Slide 2 Unit 7 Probability

3. Chapter 12 Section 1 - Slide 3 Copyright © 2009 Pearson Education, Inc. WHAT YOU WILL LEARN Empirical probability and theoretical probability Compound probability, conditional probability, and binomial probability Odds against an event and odds in favor of an event Expected value Tree diagrams

4. Chapter 12 Section 1 - Slide 4 Copyright © 2009 Pearson Education, Inc. WHAT YOU WILL LEARN Mutually exclusive events and independent events The counting principle, permutations, and combinations

5. Copyright © 2009 Pearson Education, Inc. Chapter 12 Section 1 - Slide 5 Section 1 The Nature of Probability

6. Chapter 12 Section 1 - Slide 6 Copyright © 2009 Pearson Education, Inc. Definitions An experiment is a controlled operation that yields a set of results. The possible results of an experiment are called its outcomes. An event is a subcollection of the outcomes of an experiment.

7. Chapter 12 Section 1 - Slide 7 Copyright © 2009 Pearson Education, Inc. Definitions (continued) Empirical probability is the relative frequency of occurrence of an event and is determined by actual observations of an experiment. Theoretical probability is determined through a study of the possible outcomes that can occur for the given experiment.

8. Chapter 12 Section 1 - Slide 8 Copyright © 2009 Pearson Education, Inc. Empirical Probability Example: In 100 tosses of a fair die, 19 landed showing a 3. Find the empirical probability of the die landing showing a 3. Let E be the event of the die landing showing a 3.

9. Slide 12 - 9 Copyright © 2009 Pearson Education, Inc. Of 12 children playing at the playground, 4 are playing on the swing set. Determine the empirical probability that the next child to the playground will play on the swing set. a. c. b. d.

10. Slide 12 - 10 Copyright © 2009 Pearson Education, Inc. Of 12 children playing at the playground, 4 are playing on the swing set. Determine the empirical probability that the next child to the playground will play on the swing set. a. c. b. d.

11. Chapter 12 Section 1 - Slide 11 Copyright © 2009 Pearson Education, Inc. The Law of Large Numbers The law of large numbers states that probability statements apply in practice to a large number of trials, not to a single trial. It is the relative frequency over the long run that is accurately predictable, not individual events or precise totals.

12. Copyright © 2009 Pearson Education, Inc. Chapter 12 Section 1 - Slide 12 Section 2 Theoretical Probability

13. Chapter 12 Section 1 - Slide 13 Copyright © 2009 Pearson Education, Inc. Equally likely outcomes If each outcome of an experiment has the same chance of occurring as any other outcome, they are said to be equally likely outcomes. For equally likely outcomes, the probability of Event E may be calculated with the following formula.

14. Chapter 12 Section 1 - Slide 14 Copyright © 2009 Pearson Education, Inc. Example A die is rolled. Find the probability of rolling a) a 2. b) an odd number. c) a number less than 4. d) an 8. e) a number less than 9.

15. Chapter 12 Section 1 - Slide 15 Copyright © 2009 Pearson Education, Inc. Solutions: There are six equally likely outcomes: 1, 2, 3, 4, 5, and 6. a) b) There are three ways an odd number can occur: 1, 3 or 5. c) Three numbers are less than 4.

16. Chapter 12 Section 1 - Slide 16 Copyright © 2009 Pearson Education, Inc. d) There are no outcomes that will result in an 8. e) All outcomes are less than 9. The event must occur and the probability is 1. Solutions: There are six equally likely outcomes: 1, 2, 3, 4, 5, and 6 (continued)

17. Slide 12 - 17 Copyright © 2009 Pearson Education, Inc. Each of the numbers 0-9 is written on a sheet of paper and the ten sheets of paper are placed in a box. If one sheet of paper is selected at random from the box, determine the probability that the number selected is less than 3. a. c. b. d.

18. Slide 12 - 18 Copyright © 2009 Pearson Education, Inc. Each of the numbers 0-9 is written on a sheet of paper and the ten sheets of paper are placed in a box. If one sheet of paper is selected at random from the box, determine the probability that the number selected is less than 3. a. c. b. d.

19. Chapter 12 Section 1 - Slide 19 Copyright © 2009 Pearson Education, Inc. Important Facts The probability of an event that cannot occur is 0. The probability of an event that must occur is 1. Every probability is a number between 0 and 1 inclusive; that is, 0  P(E)  1. The sum of the probabilities of all possible outcomes of an experiment is 1.

20. Chapter 12 Section 1 - Slide 20 Copyright © 2009 Pearson Education, Inc. Example A standard deck of cards is well shuffled. Find the probability that the card is selected. a) a 10. b) not a 10. c) a heart. d) an ace, 2, or 3. e) diamond and spade. f) a card greater than 4 and less than 7.

21. Chapter 12 Section 1 - Slide 21 Copyright © 2009 Pearson Education, Inc. Example (continued) a) a 10 There are four 10’s in a deck of 52 cards. b) not a 10

22. Chapter 12 Section 1 - Slide 22 Copyright © 2009 Pearson Education, Inc. Example continued c) a heart There are 13 hearts in the deck. d) an ace, 2 or 3 There are 4 aces, 4 twos and 4 threes, or a total of 12 cards.

23. Chapter 12 Section 1 - Slide 23 Copyright © 2009 Pearson Education, Inc. Example continued d) diamond and spade The word and means both events must occur. This is not possible. e) a card greater than 4 and less than 7 The cards greater than 4 and less than 7 are 5’s and 6’s (or a total of 8 cards).

24. Copyright © 2009 Pearson Education, Inc. Chapter 12 Section 1 - Slide 24 Section 3 Odds

25. Chapter 12 Section 1 - Slide 25 Copyright © 2009 Pearson Education, Inc. Odds Against

26. Chapter 12 Section 1 - Slide 26 Copyright © 2009 Pearson Education, Inc. Example: Odds Against Example: Find the odds against rolling a 5 on one roll of a die. The odds against rolling a 5 are 5:1. odds against rolling a 5

27. Chapter 12 Section 1 - Slide 27 Copyright © 2009 Pearson Education, Inc. Odds in Favor

28. Chapter 12 Section 1 - Slide 28 Copyright © 2009 Pearson Education, Inc. Example Find the odds in favor of landing on blue in one spin of the spinner. The odds in favor of spinning blue are 3:5.

29. Slide 12 - 29 Copyright © 2009 Pearson Education, Inc. Five red apples and six yellow apples are placed in a bag. If one apple is selected at random, determine the odds in favor of the apple being red. a.5:11 b.6:11 c.5:6 d.6:5

30. Slide 12 - 30 Copyright © 2009 Pearson Education, Inc. Five red apples and six yellow apples are placed in a bag. If one apple is selected at random, determine the odds in favor of the apple being red. a.5:11 b.6:11 c.5:6 d.6:5

31. Chapter 12 Section 1 - Slide 31 Copyright © 2009 Pearson Education, Inc. Probability from Odds Example: The odds against spinning a blue on a certain spinner are 4:3. Find the probability that a) a blue is spun. b) a blue is not spun.

32. Chapter 12 Section 1 - Slide 32 Copyright © 2009 Pearson Education, Inc. Solution Since the odds are 4:3 the denominators must be 4 + 3 = 7. The probabilities ratios are: