2. Chapters 14 and 15 Probability Basics Probability Fundamentals Counting Rules Applied to the Equally Likely Model

3. Birthday Problem What is the smallest number of people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2? Answer: 23 No. of people23304060 Probability.507.706.891.994

4. Probability Formal study of uncertainty The engine that drives Statistics Primary objective of Chapters 14 and 15: 1.use the rules of probability to calculate appropriate measures of uncertainty. 2.Learn the probability basics so that we can do Statistical Inference

5. Introduction Your favorite basketball team has the ball and trails by 2 points with little time remaining in the game. Should your team attempt a game- tying 2-pointer or go for a buzzer-beating 3-pointer to win the game? (This situation has often been used in Microsoft job interviews). After a touchdown should a coach kick the extra point or go for two? On 4th down should your favorite football team punt or try for the first down? With a man on first base and no one out, should the manager call for a sacrifice bunt? If your favorite basketball team has a 3 point lead with little time left on the clock and the other team has the ball, should your team foul?

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

7. Coin toss 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). First series of tosses Second series The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.

8. The Laws of Probability 1.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 Approaches to Probability

9. Approaches to Probability (cont.) 2.Subjective probability individual assigns prob. based on personal experience, anecdotal evidence, etc. 3.Classical approach every possible outcome has equal probability (more later)

10. 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

11. 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  Certain event: S

12. 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} 3.Select 1 card from a deck of 52 cards. S = {all 52 cards}

13. Laws of Probability

14. Coin Toss Example: S = {Head, Tail} Probability of heads = 0.5 Probability of tails = 0.5 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(Tail) = 0.5 Probability rules (cont’d) Venn diagram: Sample space made up of an event A and its complement A, i.e., everything that is not A.

15. Birthday Problem What is the smallest number of people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2? Answer: 23 No. of people23304060 Probability.507.706.891.994

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

17. Unions: , or Intersections: , and A  A 

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

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

20. 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)

21. 20 For any two events A and B P(A or B) = P(A) + P(B) - P(A and B) A B P(A) =6/13 P(B) =5/13 P(A and B) =3/13 A or B + _ P(A or B) = 8/13 General Addition Rule

22. Laws of Probability (cont.) Multiplication Rule 6. For two independent events A and B P(A and B) = P(A  B) = P(A) × P(B)

23. 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) 6. for two independent events A and B, P(A and B) = P(A) × P(B)

24. M&M candies ColorBrownRedYellowGreenOrangeBlue Probability0.30.2 0.1 ? 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? What is the probability that a random M&M is any of red, yellow, or orange? 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 P(red or yellow or orange) = P(red) + P(yellow) + P(orange) = 0.2 + 0.2 + 0.1 = 0.5

25. Example: college students L = {student lives on campus} M = {student purchases a meal plan} P(a student either lives or eats on campus) = P(L or M) = P(L) + P(M) - P(L and M) =0.56 + 0.62 – 0.42 = 0.76 Suppose 56% of all students live on campus, 62% of all students purchase a campus meal plan and 42% do both. Question: what is the probability that a randomly selected student either lives OR eats on campus.

26. The Equally Likely Probability Model Applications and Counting Methods

27. Assigning Probabilities zIf an experiment has N outcomes, then each outcome has probability 1/N of occurring zIf an event A 1 has n 1 outcomes, then P(A 1 ) = n 1 /N

28. Dice You toss two dice. What is the probability of the outcomes summing to 5? 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 This is S: {(1,1), (1,2), (1,3), ……etc.}

29. We Need Efficient Methods for Counting Outcomes

30. Counting in “Either-Or” Situations NCAA Basketball Tournament, 68 teams: how many ways can the “bracket” be filled out? 1.How many games? 2.2 choices for each game 3.Number of ways to fill out the bracket: 2 67 = 1.5 × 10 20 Earth pop. about 6 billion; everyone fills out 100 million different brackets Chances of getting all games correct is about 1 in 1,000

31. Counting Example zIn the knock-out stages of a soccer tournament, when a game ends in a tie the winner is determined by a penalty-kick shootout. The shootout, which consists of an alternating sequence of penalty kicks, is won by the team with the greatest goal tally after 5 kicks per team. zA coach has selected the 5 players that will take the penalty kicks in a shootout. In how many ways can the coach designate the order in which the 5 players take the penalty kicks?

32. Solution zThere are 5 players to choose to take the first penalty kick, 4 remaining players to take the second penalty kick, 3 players for the third penalty kick, 2 players for the fourth penalty kick, and 1 player for the fifth penalty kick. zThe number of possible arrangements is therefore 5  4  3  2  1 = 120

33. Efficient Methods for Counting Outcomes zFactorial Notation: n!=1  2  …  n zExamples 1!=1; 2!=1  2=2; 3!= 1  2  3=6; 4!=24; 5!=120; zSpecial definition: 0!=1

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

35. Permutations A B C D E zHow many ways can we choose 2 letters from the above 5, without replacement, when the order in which we choose the letters is important? z5  4 = 20

36. Permutations (cont.)

37. Permutations with calculator and Excel zCalculator non-graphing: nPr zGraphing p. 9 of T I Calculator Commands (math button) zExcel Insert function: Statistical, Permut

38. Combinations A B C D E zHow 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? z5  4 = 20 when order important  Divide by 2: (5  4)/2 = 10 ways

39. Combinations (cont.)

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

41. Chances of Winning?

42. Example: Illinois State Lottery

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

44. The Forrest Gump Visualization of Your Lottery Chances zHow large is 195,249,054? z$1 bill and $100 bill both 6” in length z10,560 bills = 1 mile zLet’s start with 195,249,053 $1 bills and one $100 bill … z… 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

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

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

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

51. Chances of Winning NC Powerball Lottery? zRemember: one of the bills you put down is a $100 bill; all others are $1 bills. zPut on a blindfold and begin walking along the trail of bills. zYour chance of winning the lottery is the same as your chance of selecting the $100 bill if you stop at a random location along the trail and pick up a bill.

52. End of Chapters 14 and 15 (part 1)