1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 13 From Randomness to Probability

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 2 Dealing with Random Phenomena When we toss a coin, we know that it will be either a head or a tail, but not which one. A random phenomenon is a situation in which we know what outcomes could happen, but we don’t know which particular outcome did or will happen. We use probability to assess the likelihood that a random phenomenon has a particular outcome. In each case of flipping a coin, the next outcome is uncertain, however, over the long run of many repetitions, a pattern emerges.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 3 Probability The probability of an event is its long-run relative frequency. While we may not be able to predict a particular individual outcome, we can talk about what will happen in the long run. For any random phenomenon, each attempt, or trial, generates an outcome. Something happens on each trial, and we call whatever happens the outcome. These outcomes are individual possibilities, like the number we see on top when we roll a die.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 4 Probability (cont.) Sometimes we are interested in a combination of outcomes (e.g., a die is rolled and comes up even). A combination of outcomes is called an event. When thinking about what happens with combinations of outcomes, things are simplified if the individual trials are independent. Roughly speaking, this means that the outcome of one trial doesn’t influence or change the outcome of another. For example, coin flips are independent.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Probability Experiments Probability experiment An action, or trial, through which specific results (counts, measurements, or responses) are obtained. Outcome The result of a single trial in a probability experiment. Sample Space The set of all possible outcomes of a probability experiment. Event Consists of one or more outcomes and is a subset of the sample space.

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Probability Experiments Probability experiment: Roll a die Sample space: {1, 2, 3, 4, 5, 6} Outcome: {3} Event: {Die is even}={2, 4, 6} Larson/Farber 4th ed 6

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example: Identifying the Sample Space A probability experiment consists of tossing a coin and then rolling a six-sided die. Describe the sample space. Solution: There are two possible outcomes when tossing a coin: a head (H) or a tail (T). For each of these, there are six possible outcomes when rolling a die: 1, 2, 3, 4, 5, or 6. One way to list outcomes for actions occurring in a sequence is to use a tree diagram.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution: Identifying the Sample Space Tree diagram: H1 H2 H3 H4 H5 H6 T1 T2 T3 T4 T5 T6 The sample space has 12 outcomes: {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

9. Probability We use probability to assess the likelihood that a random phenomenon has a particular outcome. If A represents a particular outcome or set of outcomes of a random phenomenon, we write P(A) to denote the probability that event A will occur. There are two requirements for a probability:  A probability is a number between 0 and 1 and can be expressed as fractions, decimals or per cents.  For any event A, 0 ≤ P(A) ≤ 1. Larson/Farber 4th ed 9

10. Probability Rules 1. Range of probabilities rule The probability of an event E is between 0 and 1, inclusive. 0 ≤ P(E) ≤ 1 Larson/Farber 4th ed 10 [ ] 00.51 ImpossibleUnlikely Even chance LikelyCertain

11. Slide 14- 11 Probability 2.“Something has to happen rule”:  The probability of the set of all possible outcomes of a trial must be 1.  P(S) = 1 (S represents the entire sample space or the set of all possible outcomes.)

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 12 The Law of Large Numbers Consider flipping a fair coin many, many times, say 1000. Approximately how many heads do you expect to see? The Law of Large Numbers (LLN) says that the long-run relative frequency of repeated independent events gets closer and closer to the true relative frequency as the number of trials increases.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Law of Large Numbers As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of the event.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 14 The Law of Large Numbers (cont.) Next consider flipping a fair coin, if heads comes up on each of the first 10 flips, what do you think the chance is that tails will come up on the next flip? The common misunderstanding of the LLN is that random phenomena are supposed to compensate some for whatever happened in the past. This is just not true!!!

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 15 Probability Thanks to the LLN, we know that relative frequencies settle down in the long run, so we can officially give the name probability to that value. BUT REMEMBER: Probabilities must be between 0 and 1, inclusive. A probability of 0 indicates impossibility. A probability of 1 indicates certainty.

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 14- 16 Equally Likely Outcomes When probability was first studied, a group of French mathematicians looked at games of chance in which all the possible outcomes were equally likely. It’s equally likely to get any one of six outcomes from the roll of a fair die. It’s equally likely to get heads or tails from the toss of a fair coin. However, keep in mind that events are not always equally likely. A skilled basketball player has a better than 50-50 chance of making a free throw.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Types of Probability Classical (theoretical) Probability Each outcome in a sample space is equally likely. /

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example: Finding Classical Probabilities 1. Event A: rolling a 3 2. Event B: rolling a 7 3. Event C: rolling a number less than 5 You roll a six-sided die. Find the probability of each event.

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Types of Probability Empirical (statistical) Probability Based on observations obtained from probability experiments. Relative frequency of an event.

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example: Finding Empirical Probabilities A company is conducting an online survey of randomly selected individuals to determine if traffic congestion is a problem in their community. So far, 320 people have responded to the survey. What is the probability that the next person that responds to the survey says that traffic congestion is a serious problem in their community? ResponseNumber of times, f Serious problem123 Moderate problem115 Not a problem 82 Σf = 320

21. PROBABILITY BASED ON INTUITION Suppose a sports reporter predicts that the Yankees have a 75% chance of beating the Red Sox in their next game. How did he arrive at this value? The reporter is giving his or her professional assessment based on their knowledge of the players and past experiences. We classify this type of probability assignments as Informed Intuition Larson/Farber 4th ed 21

22. WAYS OF COUNTING Objectives Determine the number of ways a group of objects can be arranged in order Determine the number of ways to choose several objects from a group without regard to order Use the counting principles to find probabilities Larson/Farber 4th ed 22

23. Fundamental Counting Principle If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m*n. Can be extended for any number of events occurring in sequence.

24. Example: Fundamental Counting Principle You are purchasing a new car. The possible manufacturers, car sizes, and colors are listed. Manufacturer: Ford, GM, Honda Car size: compact, midsize Color: white (W), red (R), black (B), green (G) How many different ways can you select one manufacturer, one car size, and one color? Larson/Farber 4th ed 24

25. Permutations CONSIDER: IN HOW MANY WAYS CAN YOU ARRANGE THE FOLLOWING 3 BOOKS ON A SHELF? A MATH BOOK, A HISTORY BOOK AND A SCIENCE BOOK Larson/Farber 4th ed 25

26. PERMUTATION An ordered arrangement of objects The number of different permutations of n distinct objects is n! (n factorial)  n! = n∙(n – 1)∙(n – 2)∙(n – 3)∙ ∙ ∙3∙2 ∙1  0! = 1  Examples: 6! = 6∙5∙4∙3∙2∙1 = 720 4! = 4∙3∙2∙1 = 24

27. Example: Permutation of n Objects The objective of a 9 x 9 Sudoku number puzzle is to fill the grid so that each row, each column, and each 3 x 3 grid contain the digits 1 to 9. How many different ways can the first row of a blank 9 x 9 Sudoku grid be filled? Larson/Farber 4th ed 27 Solution: The number of permutations is 9!= 9∙8∙7∙6∙5∙4∙3∙2∙1 = 362,880 ways

28. Permutations Permutation of n objects taken r at a time The number of different permutations of n distinct objects taken r at a time Larson/Farber 4th ed 28 ■ where r ≤ n

29. Example: Finding nPr Find the number of ways of forming three-digit codes in which no digit is repeated. Larson/Farber 4th ed 29 Solution: You need to select 3 digits from a group of 10 n = 10, r = 3

30. Example: Finding nPr Forty-three race cars started the 2007 Daytona 500. How many ways can the cars finish first, second, and third? Larson/Farber 4th ed 30 Solution: You need to select 3 cars from a group of 43 n = 43, r = 3

31. Distinguishable Permutations The number of distinguishable permutations of n objects where n 1 are of one type, n 2 are of another type, and so on Larson/Farber 4th ed 31 ■ where n 1 + n 2 + n 3 +∙∙∙+ n k = n

32. Example: Distinguishable Permutations A building contractor is planning to develop a subdivision that consists of 6 one-story houses, 4 two- story houses, and 2 split-level houses. In how many distinguishable ways can the houses be arranged? Larson/Farber 4th ed 32 Solution: There are 12 houses in the subdivision n = 12, n 1 = 6, n 2 = 4, n 3 = 2

33. NEXT CONSIDER: IN HOW MANY WAYS CAN YOU CHOOSE THREE PEOPLE FROM THIS CLASS TO FORM A COMMITTEE? ASK YOURSELF: IS ORDER IMPORTANT?????

34. Combinations Combination of n objects taken r at a time A selection of r objects from a group of n objects without regard to order Larson/Farber 4th ed 34 ■

35. Example: Combinations A state’s department of transportation plans to develop a new section of interstate highway and receives 16 bids for the project. The state plans to hire four of the bidding companies. How many different combinations of four companies can be selected from the 16 bidding companies? Larson/Farber 4th ed 35 Solution: You need to select 4 companies from a group of 16 n = 16, r = 4 Order is not important

36. Solution: Combinations Larson/Farber 4th ed 36

37. Example: Probability Using the Fundamental Counting Principle Your college identification number consists of 8 digits. Each digit can be 0 through 9 and each digit can be repeated. What is the probability of getting your college identification number when randomly generating eight digits? Larson/Farber 4th ed 37

38. Solution: Probability Using the Fundamental Counting Principle Each digit can be repeated There are 10 choices for each of the 8 digits Using the Fundamental Counting Principle, there are 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 = 10 8 = 100,000,000 possible identification numbers Only one of those numbers corresponds to your ID number Larson/Farber 4th ed 38 P(your ID number) =

39. Example: Finding Probabilities A student advisory board consists of 17 members. Three members serve as the board’s chair, secretary, and webmaster. Each member is equally likely to serve any of the positions. What is the probability of selecting at random the three members that hold each position? Larson/Farber 4th ed 39

40. Solution: Finding Probabilities There is only one favorable outcome There are ways the three positions can be filled Larson/Farber 4th ed 40

41. Example: Finding Probabilities You have 11 letters consisting of one M, four Is, four Ss, and two Ps. If the letters are randomly arranged in order, what is the probability that the arrangement spells the word Mississippi? Larson/Farber 4th ed 41

42. Solution: Finding Probabilities There is only one favorable outcome There are distinguishable permutations of the given letters Larson/Farber 4th ed 42 11 letters with 1,4,4, and 2 like letters

43. Example: Finding Probabilities A food manufacturer is analyzing a sample of 400 corn kernels for the presence of a toxin. In this sample, three kernels have dangerously high levels of the toxin. If four kernels are randomly selected from the sample, what is the probability that exactly one kernel contains a dangerously high level of the toxin? Larson/Farber 4th ed 43

44. Solution: Finding Probabilities The possible number of ways of choosing one toxic kernel out of three toxic kernels is 3 C 1 = 3 The possible number of ways of choosing three nontoxic kernels from 397 nontoxic kernels is 397 C 3 = 10,349,790 Using the Multiplication Rule, the number of ways of choosing one toxic kernel and three nontoxic kernels is 3 C 1 ∙ 397 C 3 = 3 ∙ 10,349,790 3 = 31,049,370 Larson/Farber 4th ed 44

45. Solution: Finding Probabilities The number of possible ways of choosing 4 kernels from 400 kernels is 400 C 4 = 1,050,739,900 The probability of selecting exactly 1 toxic kernel is Larson/Farber 4th ed 45

46. Section Summary Determined the number of ways a group of objects can be arranged in order Determined the number of ways to choose several objects from a group without regard to order Used the counting principles to find probabilities Larson/Farber 4th ed 46