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Practice

Practice. You bought a ticket for a fire department lottery and your brother has bought two tickets. You just read that 1000 tickets were sold. a) What is the probability you will win the grand prize? b) What is the probability that your brother will win?

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Practice

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  1. Practice • You bought a ticket for a fire department lottery and your brother has bought two tickets. You just read that 1000 tickets were sold. • a) What is the probability you will win the grand prize? • b) What is the probability that your brother will win? • c) What is the probably that you or your bother will win?

  2. 5.2 • A) 1/1000 = .001 • B)2/1000 = .002 • C) .001 + .002 = .003

  3. Practice • Assume the same situation at before except only a total of 10 tickets were sold and there are two prizes. • a) Given that you didn’t win the first prize, what is the probability you will win the second prize? • b) What is the probability that your borther will win the first prize and you will win the second prize? • c) What is the probability that you will win the first prize and your brother will win the second prize? • d) What is the probability that the two of you will win the first and second prizes?

  4. 5.3 • A) 1/9 = .111 • B) 2/10 * 1/9 = (.20)*(.111) = .022 • C) 1/10 * 2/9 = (.10)*(.22) = .022 • D) .022 + .022 = .044

  5. Practice • In some homes a mother’s behavior seems to be independent of her baby's, and vice versa. If the mother looks at her child a total of 2 hours each day, and the baby looks at the mother a total of 3 hours each day, and if they really do behave independently, what is the probability that they will look at each other at the same time?

  6. 5.8 • 2/24 = .083 • 3/24 = .125 • .083*.125 = .01

  7. Practice • Abe ice-cream shot has six different flavors of ice cream, and you can order any combination of any number of them (but only one scoop of each flavor). How many different ice-cream cone combinations could they truthfully advertise (note, we don’t care about the order of the scoops and an empty cone doesn’t count).

  8. 5.29 6 + 15 + 20 +15 + 6 + 1 = 63

  9. Suppose you live in a place that has a constant chance of being struck by lightning at any time through the year. Suppose that the strikes are random: every day the chance of a strike is the same, and the rate works out to one strike a month. Your house is hit by lightning today, Monday. What is the most likely day for the next bolt to strike your house?

  10. Tuesday! • Say the prob of lighting hitting is .025 • On Tues prob of hitting is .025 • For Wed to be next hit this would have to be true: no hit on Tues AND hit on Wed 975*.025=.0243 • For Thursday .975*.975*.025=.0237

  11. Remember • Playing perfect black jack – the probability of winning a hand is .498 • What is the probability that you will win 8 of the next 10 games of blackjack?

  12. Binomial Distribution Ingredients: N = total number of events p = the probability of a success on any one trial q = (1 – p) = the probability of a failure on any one trial X = number of successful events

  13. Binomial Distribution Ingredients: N = total number of events p = the probability of a success on any one trial q = (1 – p) = the probability of a failure on any one trial X = number of successful events

  14. Binomial Distribution Ingredients: N = total number of events p = the probability of a success on any one trial q = (1 – p) = the probability of a failure on any one trial X = number of successful events p = .0429

  15. Binomial Distribution • What if you are interested in the probability of winning at least 8 games of black jack? • To do this you need to know the distribution of these probabilities

  16. Probability of Winning Blackjack • p = .498, N = 10

  17. Probability of Winning Blackjack • p = .498, N = 10

  18. Probability of Winning Blackjack • p = .498, N = 10

  19. Probability of Winning Blackjack • p = .498, N = 10

  20. Binomial Distribution p Games Won

  21. Hypothesis Testing • You wonder if winning at least 7 games of blackjack is significantly (.05) better than what would be expected due to chance. • H1= Games won > 6 • H0= Games won < or equal to 6 • What is the probability of winning 7 or more games?

  22. Binomial Distribution p Games Won

  23. Binomial Distribution p Games Won

  24. Probability of Winning Blackjack • p = .498, N = 10

  25. Probability of Winning Blackjack • p = .498, N = 10 • p of winning 7 or more games • .115+.044+.009+.001 = .169 • p > .05 • Not better than chance

  26. Practice • The probability at winning the “Statistical Slot Machine” is .08. • Create a distribution of probabilities when N = 10 • Determine if winning at least 4 games of slots is significantly (.05) better than what would be expected due to chance.

  27. Probability of Winning Slot

  28. Binomial Distribution p Games Won

  29. Probability of Winning Slot • p of winning at least 4 games • .005+.001+.000 . . . .000 = .006 • p< .05 • Winning at least 4 games is significantly better than chance

  30. Binomial Distribution • These distributions can be described with means and SD. • Mean = Np • SD =

  31. Binomial Distribution • Black Jack; p = .498, N =10 • M = 4.98 • SD = 1.59

  32. Binomial Distribution p Games Won

  33. Binomial Distribution • Statistical Slot Machine; p = .08, N = 10 • M = .8 • SD = .86

  34. Binomial Distribution Note: as N gets bigger, distributions will approach normal p Games Won

  35. Next Step • You think someone is cheating at BLINGOO! • p = .30 of winning • You watch a person play 89 games of blingoo and wins 39 times (i.e., 44%). • Is this significantly bigger than .30 to assume that he is cheating?

  36. Hypothesis • H1= .44 > .30 • H0= .44 < or equal to .30 • Or • H1= 39 wins > 26.7 wins • H0= 39 wins < or equal to 26.7 wins

  37. Distribution • Mean = 26.7 • SD = 4.32 • X = 39

  38. Z-score

  39. Results • (39 – 26.7) / 4.32 = 2.85 • p = .0021 • p < .05 • .44 is significantly bigger than .30. There is reason to believe the person is cheating! • Or – 39 wins is significantly more than 26.7 wins (which are what is expected due to chance)

  40. BLINGOO Competition • You and your friend enter at competition with 2,642 other players • p = .30 • You win 57 of the 150 games and your friend won 39. • Afterward you wonder how many people • A) did better than you? • B) did worse than you? • C) won between 39 and 57 games • You also wonder how many games you needed to win in order to be in the top 10%

  41. Blingoo • M = 45 • SD = 5.61 • A) did better than you? • (57 – 45) / 5.61 = 2.14 • p = .0162 • 2,642 * .0162 = 42.8 or 43 people

  42. Blingoo • M = 45 • SD = 5.61 • A) did worse than you? • (57 – 45) / 5.61 = 2.14 • p = .9838 • 2,642 * .9838 = 2,599.2 or 2,599 people

  43. Blingoo • M = 45 • SD = 5.61 • A) won between 39 and 57 games? • (57 – 45) / 5.61 = 2.14 ; p = .4838 • (39 – 45) / 5.61 = -1.07 ; p = .3577 • .4838 + .3577 = .8415 • 2,642 * .8415 = 2,223.2 or 2, 223 people

  44. Blingoo • M = 45 • SD = 5.61 • You also wonder how many games you needed to win in order to be in the top 10% • Z = 1.28 • 45 + 5.61 (1.28) = 52.18 games or 52 games

  45. Is a persons’ size related to if they were bullied • You gathered data from 209 children at Springfield Elementary School. • Assessed: • Height (short vs. not short) • Bullied (yes vs. no)

  46. Results Ever Bullied

  47. Results Ever Bullied

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