Chapters 14 and 15

1. Chapters 14 and 15 Probability and counting

2. 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 We will solve this problem a few slides later using the laws of probability

3. Probability Formal study of uncertainty The engine that drives Statistics Primary objectives: use the rules of probability to calculate appropriate measures of uncertainty. Learn the probability basics so that we can do Statistical Inference

4. 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 or the birthday problem

5. Randomness and probability Randomness ≠ chaos A phenomenon is random if individual outcomes are uncertain, but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

6. The result of any single coin toss is random. But the result over many tosses is predictable, as long as the trials are independent (i.e., the outcome of a new coin flip is not influenced by the result of the previous flip). Coin toss The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials. First series of tosses Second series

7. 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

8. 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)

9. 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

10. 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. Null event: the empty set F Certain event: S

11. Examples 1. Toss a coin once S = {H, T}; A = {H}, B = {T} 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} • Select 1 card from a deck of 52 cards. S = {all 52 cards}

12. Laws of Probability

13. Coin Toss Example: S ={Head, Tail} Probability of heads = 0.5 Probability of tails = 0.5 Probability rules (cont’d) 3) The complement of any event A is the event that A does not occur, written as A. The complement rule states that the probability of an event not occurring is 1 minus the probability that is does occur. P(not A) = P(A) = 1 − P(A) Tail  = not Tail = Head P(Tail ) = 1 − P(Head) = 0.5  Venn diagram: Sample space made up of an event A and its complementary A , i.e., everything that is not A.

14. 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

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

16. Unions: , orIntersections: , and AÇB AÈB

17. Mutually Exclusive (Disjoint) Events Venn Diagrams A and B disjoint: A  B= • Mutually exclusive or disjoint events-no outcomes from S in common AÇB AÈB A and B not disjoint

18. Addition Rule for Disjoint Events 4. If A and B are disjoint events, then P(A or B) = P(A) + P(B)

19. Laws of Probability (cont.) General Addition Rule 5. For any two events A and B P(A or B) = P(A) + P(B) – P(A and B)

20. A or B General Addition Rule For any two events A and B P(A or B) = P(A) + P(B) - P(A and B) P(A) =6/13 A + P(B) =5/13 _ B P(A and B) =3/13 P(A or B) = 8/13

21. 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 or B) = P(A) + P(B) • 5. For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B)

22. M&M candies If you draw an M&M candy at random from a bag, the candy will have one of six colors. The probability of drawing each color depends on the proportions manufactured, as described here: What is the probability that an M&M chosen at random is blue? S = {brown, red, yellow, green, orange, blue} P(S) = P(brown) + P(red) + P(yellow) + P(green) + P(orange) + P(blue) = 1 P(blue) = 1 – [P(brown) + P(red) + P(yellow) + P(green) + P(orange)] = 1 – [0.3 + 0.2 + 0.2 + 0.1 + 0.1] = 0.1 What is the probability that a random M&M is any of red, yellow, or orange? P(red or yellow or orange) = P(red) + P(yellow) + P(orange) = 0.2 + 0.2 + 0.1 = 0.5

23. 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 orB) = P(A) + P(B) - P(A andB) =P({2, 4, 6}) + P({1, 2, 3}) - P({2}) = 3/6 + 3/6 - 1/6 = 5/6

24. THE RELATIONSHIP BETWEEN ODDS AND PROBABILITIES World Series Odds The odds at the above link are the odds against a team winning the World Series, though the author claims they’re “odds for winning the World Series” Odds are frequently a source of confusion. Odds for? Odds against? From probability to odds From odds to probability

25. If event A has probability P(A), then the odds in favor of A are P(A) to 1-P(A). It follows that the odds against A are 1-P(A) to P(A) If the probability of an earthquake in California is .25, then the odds in favor of an earthquake are .25 to .75 or 1 to 3. The odds against an earthquake are .75 to .25 or 3 to 1 From Probability to Odds

26. If the odds in favor of an event E are a to b, then P(E)=a/(a+b) in addition, P(E’)=b/(a+b) If the odds in favor of UNC winning the NCAA’s are 3 (a) to 1 (b), then P(UNC wins)=3/4 P(UNC does not win)= 1/4 From Odds to Probability

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

28. 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

29. Dice You toss two dice. What is the probability of the outcomes summing to 5? This isS: {(1,1), (1,2), (1,3), ……etc.} There are 36 possible outcomes in S, all equally likely (given fair dice). Thus, the probability of any one of them is 1/36. P(the roll of two dice sums to 5) = P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 / 36 = 0.111

30. We Need Efficient Methods for Counting Outcomes

31. 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?

32. 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)

33. 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

34. 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.

35. 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

36. 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?

37. 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

38. 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

39. Factorials with calculators and Excel • Calculator: non-graphing: x ! (second function) graphing: bottom p. 9 T I Calculator Commands (math button) • Excel: Insert function: Math and Trig category, FACT function

40. 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 • 20! is roughly the age of the universe in seconds

41. 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

42. Permutations (cont.)

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

44. 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

45. Combinations (cont.)

46. ST 311 Powerball Lottery From the numbers 1 through 20, choose 6 different numbers. Write them on a piece of paper.

47. And the numbers are ... • 16 • 11 • 2 • 10 • 8 • 4 wow scream

48. Chances of Winning?

49. Example: Illinois State Lottery

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