Probability

1. Probability Formal study of uncertainty The engine that drives statistics

2. Introduction • Nothing in life is certain • We gauge the chances of successful outcomes in business, medicine, weather, and other everyday situations such as the lottery (recall the birthday problem)

3. History • For most of human history, probability, the formal study of the laws of chance, has been used for only one thing: gambling

4. History (cont.) • Nobody knows exactly when gambling began; goes back at least as far as ancient Egypt where 4-sided “astragali” (made from animal heelbones) were used

5. Rule 1: Let Caesar win IV out of V times History (cont.) • The Roman emperor Claudius (10BC-54AD) wrote the first known treatise on gambling. • The book “How to Win at Gambling” was lost.

6. Approaches to Probability • Relative frequency event probability = x/n, where x=# of occurrences of event of interest, n=total # of observations • Coin, die tossing; nuclear power plants? • Limitations repeated observations not practical

7. Approaches to Probability (cont.) • Subjective probability individual assigns prob. based on personal experience, anecdotal evidence, etc. • Classical approach every possible outcome has equal probability (more later)

8. Basic Definitions • Experiment: act or process that leads to a single outcome that cannot be predicted with certainty • Examples: 1. Toss a coin 2. Draw 1 card from a standard deck of cards 3. Arrival time of flight from Atlanta to RDU

9. Basic Definitions (cont.) • Sample space: all possible outcomes of an experiment. Denoted by S • Event: any subset of the sample space S; typically denoted A, B, C, etc. Simple event: event with only 1 outcome Null event: the empty set F Certain event: S

10. Examples 1. Toss a coin once S = {H, T}; A = {H}, B = {T} simple events 2. Toss a die once; count dots on upper face S = {1, 2, 3, 4, 5, 6} A=even # of dots on upper face={2, 4, 6} B=3 or fewer dots on upper face={1, 2, 3}

11. Laws of Probability

12. Laws of Probability (cont.) 3. P(A’ ) = 1 - P(A) For an event A, A’ is the complement of A; A’ is everything in S that is not in A. S A' A

13. Birthday Problem • What is the smallest number of people you need in a group so that the probability of 2 or morepeople having the same birthday is greater than 1/2? • Answer: 23 No. of people 23 30 40 60 Probability .507 .706 .891 .994

14. Example: Birthday Problem • A={at least 2 people in the group have a common birthday} • A’ = {no one has common birthday}

15. Unions and Intersections S AÇB A B AÈB

16. Mutually Exclusive Events • Mutually exclusive events-no outcomes from S in common A Ç B = Æ S A B

17. Laws of Probability (cont.) Addition Rule for Disjoint Events: 4. If A and B are disjoint events, then P(A  B) = P(A) + P(B)

18. 5. For two independent events A and B P(A  B) = P(A) × P(B)

19. Laws of Probability (cont.) General Addition Rule 6. For any two events A and B P(A  B) = P(A) + P(B) – P(A  B)

20. P(AÈB)=P(A) + P(B) - P(A Ç B) S AÇB A B

21. Example: toss a fair die once • S = {1, 2, 3, 4, 5, 6} • A = even # appears = {2, 4, 6} • B = 3 or fewer = {1, 2, 3} • P(A È B) = P(A) + P(B) - P(A Ç B) =P({2, 4, 6}) + P({1, 2, 3}) - P({2}) = 3/6 + 3/6 - 1/6 = 5/6

22. Laws of Probability: Summary • 1. 0  P(A)  1 for any event A • 2. P() = 0, P(S) = 1 • 3. P(A’) = 1 – P(A) • 4. If A and B are disjoint events, then P(A  B) = P(A) + P(B) • 5. If A and B are independent events, then P(A  B) = P(A) × P(B) • 6. For any two events A and B, P(A  B) = P(A) + P(B) – P(A  B)

23. Probability Models The Equally Likely Approach (also called the Classical Approach)

24. Assigning Probabilities • If an experiment has N outcomes, then each outcome has probability 1/N of occurring • If an event A1 has n1 outcomes, then P(A1) = n1/N

25. We Need Efficient Methods for Counting Outcomes

26. Product Rule for Ordered Pairs • A student wishes to commute to a junior college for 2 years and then commute to a state college for 2 years. Within commuting distance there are 4 junior colleges and 3 state colleges. How many junior college-state college pairs are available to her?

27. Product Rule for Ordered Pairs • junior colleges: 1, 2, 3, 4 • state colleges a, b, c • possible pairs: (1, a) (1, b) (1, c) (2, a) (2, b) (2, c) (3, a) (3, b) (3, c) (4, a) (4, b) (4, c)

28. Product Rule for Ordered Pairs • junior colleges: 1, 2, 3, 4 • state colleges a, b, c • possible pairs: (1, a) (1, b) (1, c) (2, a) (2, b) (2, c) (3, a) (3, b) (3, c) (4, a) (4, b) (4, c) 4 junior colleges 3 state colleges total number of possible pairs = 4 x 3 = 12

29. Product Rule for Ordered Pairs • junior colleges: 1, 2, 3, 4 • state colleges a, b, c • possible pairs: (1, a) (1, b) (1, c) (2, a) (2, b) (2, c) (3, a) (3, b) (3, c) (4, a) (4, b) (4, c) In general, if there are n1 ways to choose the first element of the pair, and n2 ways to choose the second element, then the number of possible pairs is n1n2. Here n1 = 4, n2 = 3.

30. Counting in “Either-Or” Situations • NCAA Basketball Tournament: how many ways can the “bracket” be filled out? • How many games? • 2 choices for each game • Number of ways to fill out the bracket: 263 = 9.2 × 1018 • Earth pop. about 6 billion; everyone fills out 1 million different brackets • Chances of getting all games correct is about 1 in 1,000

31. Counting Example • Pollsters minimize lead-in effect by rearranging the order of the questions on a survey • If Gallup has a 5-question survey, how many different versions of the survey are required if all possible arrangements of the questions are included?

32. Solution • There are 5 possible choices for the first question, 4 remaining questions for the second question, 3 choices for the third question, 2 choices for the fourth question, and 1 choice for the fifth question. • The number of possible arrangements is therefore 5  4  3  2  1 = 120

33. Efficient Methods for Counting Outcomes • Factorial Notation: n!=12 … n • Examples 1!=1; 2!=12=2; 3!= 123=6; 4!=24; 5!=120; • Special definition: 0!=1

34. Factorials with calculators and Excel • Calculator: non-graphing: x ! (second function) graphing: bottom p. 9 T I Calculator Commands (math button) • Excel: Paste: math, fact

35. Factorial Examples • 20! = 2.43 x 1018 • 1,000,000 seconds? • About 11.5 days • 1,000,000,000 seconds? • About 31 years • 31 years = 109 seconds • 1018 = 109 x 109 • 31 x 109 years = 109 x 109 = 1018 seconds • 20! is roughly the age of the universe in seconds

36. Permutations A B C D E • How many ways can we choose 2 letters from the above 5, without replacement, when the order in which we choose the letters is important? • 5 4 = 20

37. Permutations (cont.)

38. Permutations with calculator and Excel • Calculator non-graphing: nPr • Graphing p. 9 of T I Calculator Commands (math button) • Excel Paste: Statistical, Permut

39. Combinations A B C D E • How many ways can we choose 2 letters from the above 5, without replacement, when the order in which we choose the letters is not important? • 5 4 = 20 when order important • Divide by 2: (5  4)/2 = 10 ways

40. Combinations (cont.)

41. ST 101 Powerball Lottery From the numbers 1 through 20, choose 6 different numbers. Write them on a piece of paper.

42. Chances of Winning?

43. North Carolina Powerball Lottery Prior to Jan. 1, 2009 After Jan. 1, 2009

44. Visualize Your Lottery Chances • How large is 195,249,054? • $1 bill and $100 bill both 6” in length • 10,560 bills = 1 mile • Let’s start with 195,249,053 $1 bills and one $100 bill … • … and take a long walk, putting down bills end-to-end as we go

45. Raleigh to Ft. Lauderdale… … still plenty of bills remaining, so continue from …

46. … Ft. Lauderdale to San Diego … still plenty of bills remaining, so continue from…

47. … San Diego to Seattle … still plenty of bills remaining, so continue from …

48. … Seattle to New York … still plenty of bills remaining, so continue from …

49. … New York back to Raleigh … still plenty of bills remaining, so …

50. Go around again! Lay a second path of bills Still have ~ 5,000 bills left!!