Understanding Probabilities: Examples and Practice

2. Probabilities • What is the probability that among 23 people (this class) there will be a shared birthday?

4. The number of people required so that the probability that some pair will have a birthday separated by k days or fewer will be higher than 50% is:

5. Patrick's Casino

7. What is the probability of picking an ace?

8. Probability =

9. What is the probability of picking an ace? 4 / 52 = .077 or 7.7 chances in 100

10. Every card has the same probability of being picked

11. What is the probability of getting a 10, J, Q, or K?

12. (.077) + (.077) + (.077) + (.077) = .308 16 / 52 = .308

13. What is the probability of getting a 2 and then after replacing the card getting a 3 ?

14. (.077) * (.077) = .0059

15. What is the probability that the two cards you draw will be a black jack?

16. 10 Card = (.077) + (.077) + (.077) + (.077) = .308 Ace after one card is removed = 4/51 = .078 (.308)*(.078) = .024 But you could also get an Ace (.077) and then a ten (.078*4 = .312) or .077*.312 = .024 So the prob of either of these occurring and getting blackjack is .024 + .024 = .048

17. Practice • What is the probability of rolling a “1” using a six sided dice? • What is the probability of rolling either a “1” or a “2” with a six sided dice? • What is the probability of rolling two “1’s” using two six sided dice?

18. Practice • What is the probability of rolling a “1” using a six sided dice? 1 / 6 = .166 • What is the probability of rolling either a “1” or a “2” with a six sided dice? • What is the probability of rolling two “1’s” using two six sided dice?

19. Practice • What is the probability of rolling a “1” using a six sided dice? 1 / 6 = .166 • What is the probability of rolling either a “1” or a “2” with a six sided dice? (.166) + (.166) = .332 • What is the probability of rolling two “1’s” using two six sided dice?

20. Practice • What is the probability of rolling a “1” using a six sided dice? 1 / 6 = .166 • What is the probability of rolling either a “1” or a “2” with a six sided dice? (.166) + (.166) = .332 • What is the probability of rolling two “1’s” using two six sided dice? (.166)(.166) = .028

21. Cards • What is the probability of drawing an ace? • What is the probability your first 2 cards are aces? • What is the probability your first 4 cards are aces? • What is the probability that out of 4 cards, at least one will be an ace?

22. Cards • What is the probability of drawing an ace? • 4/52 = .0769 • What is the probability of drawing another ace? • 4/52 = .0769; 3/51 = .0588; .0769*.0588 = .0045 • What is the probability the next four cards you draw will each be an ace? • .0769*.0588*.04*.02 = .000003 • What is the probability that an ace will be in the first four cards dealt? • .0769+.078+.08+.082 = .3169

23. Probability .00 1.00 Event must occur Event will not occur

24. Probability • In this chapter we deal with discreet variables • i.e., a variable that has a limited number of values • Previously we discussed the probability of continuous variables (Z –scores) • It does not make sense to seek the probability of a single score for a continuous variable • Seek the probability of a range of scores

25. Key Terms • Independent event • When the occurrence of one event has no effect on the occurrence of another event • e.g., voting behavior, IQ, etc. • Mutually exclusive • When the occurrence of one even precludes the occurrence of another event • e.g., your year in the program, if you are in prosem

26. Key Terms • Joint probability • The probability of the co-occurrence of two or more events • The probability of rolling a one and a six • p (1, 6) • p (Blond, Blue)

27. Key Terms • Conditional probabilities • The probability that one event will occur given that some other vent has occurred • e.g., what is the probability a person will get into a PhD program given that they attended Villanova • p(Phd l Villa) • e.g., what is the probability that a person will be a millionaire given that they attended college • p($$ l college)

28. Example

29. What is the simple probability that a person will own a video game?

30. What is the simple probability that a person will own a video game? 35 / 100 = .35

31. What is the conditional probability of a person owning a video game given that he or she has children? p (video l child)

32. What is the conditional probability of a person owning a video game given that he or she has children?25 / 55 = .45

33. What is the joint probability that a person will own a video game and have children? p(video, child)

34. What is the joint probability that a person will own a video game and have children? 25 / 100 = .25

35. 25 / 100 = .25.35 * .55 = .19

36. The multiplication rule assumes that the two events are independent of each other – it does not work when there is a relationship!

37. Practice

38. p (republican) p(female)p (republican, male) p(female, republican)p (republican l male) p(male l republican)

39. p (republican) = 70 / 162 = .43p (republican, male) = 52 / 162 = .32p (republican l male) = 52 / 79 = .66

40. p(female) = 83 / 162 = .51p(female, republican) = 18 / 162 = .11p(male l republican) = 52 / 70 = .74

41. Foot Race • Three different people enter a “foot race” • A, B, C • How many different combinations are there for these people to finish?

42. Foot Race A, B, C A, C, B B, A, C B, C, A C, B, A C, A, B 6 different permutations of these three names taken three at a time

43. Foot Race • Six different people enter a “foot race” • A, B, C, D, E, F • How many different permutations are there for these people to finish?

44. Permutation Ingredients: N = total number of events r = number of events selected

45. Permutation Ingredients: N = total number of events r = number of events selected A, B, C, D, E, F Note: 0! = 1

46. Foot Race • Six different people enter a “foot race” • A, B, C, D, E, F • How many different permutations are there for these people to finish in the top three? • A, B, C A, C, D A, D, E B, C, A

47. Permutation Ingredients: N = total number of events r = number of events selected

48. Permutation Ingredients: N = total number of events r = number of events selected

49. Foot Race • Six different people enter a “foot race” • If a person only needs to finish in the top three to qualify for the next race (i.e., we don’t care about the order) how many different outcomes are there?

50. Combinations Ingredients: N = total number of events r = number of events selected