1. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Understandable Statistics Seventh Edition By Brase and Brase Prepared by: Mistah Flynn BHS Chapter Six Probability Distributions and Binomial Distributions

2. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 Statistical Experiment A statistical experiment or observation is any process by which an measurements are obtained

3. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3 Examples of Statistical Experiments Counting the number of books in the College Library Counting the number of mistakes on a page of text Measuring the amount of rainfall in your state during the month of June Counting the number of siblings you have “Start of the Year” data????

4. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 4 Random Variable a quantitative variable that assumes a value determined by chance

5. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 5 Discrete Random Variable A discrete random variable is a quantitative random variable that can take on only a finite number of values or a countable number of values. Example: the number of books in the College Library

6. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 6 Continuous Random Variable A continuous random variable is a quantitative random variable that can take on any of the countless number of values in a line interval. Example: the amount of rainfall in your state during the month of June

7. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 7 Random Variables Question about…Random Variable Let x = Type? Family x = Number of dependents in size family reported on tax return Distance from x = Distance in miles from home to store home to the store site Own dog x = 1 if own no pet; or cat = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s) Discrete Continuous Discrete

8. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 8 Probability Distribution an assignment of probabilities to the specific values of the random variable or to a range of values of the random variable

9. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9 Probability Distribution of a Discrete Random Variable A probability is assigned to each value of the random variable. The sum of these probabilities must be 1.

10. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 10 Probability distribution for the rolling of an ordinary die xP(x) 1 2 3 4 5 6

11. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 Features of a Probability Distribution xP(x) 1 2 3 4 5 6 Probabilities must be between zero and one (inclusive)  P(x) =1

12. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 Probability Histogram P(x) 1 2 3 4 5 6 ||||||||||||||

13. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 13 Mean and standard deviation of a discrete probability distribution Mean =  = expectation or expected value, the long-run average Formula :  =  x P(x)

14. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 Standard Deviation – or tolerance - σ

15. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 15 Finding the mean: xP(x) x P(x) 0.3 1.3 2.2 3.1 4.1 0.3.4.3.4 1.4  =  x P(x) = 1.4

16. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 Finding the standard deviation xP(x) x –  ( x –  ) 2 ( x –  ) 2 P(x) 0.3 1.3 2.2 3.1 4.1 – 1.4 – 0.4.6 1.6 2.6 1.96 0.16 0.36 2.56 6.76.588.048.072.256.676 1.64

17. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 17 Standard Deviation 1.28

18. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 Features of a Probability Distribution xP(x) 1 2 3 4 5 6 Probabilities must be between zero and one (inclusive)  P(x) =1

19. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 Finding the mean a die roll: xP(x) x P(x) 1.167 2.167 3.167 4.167 5.167 6.167.167.333.5.667.833 1.000 3.500  =  x P(x) = 3.5

20. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 20 Finding the standard deviation xP(x) x –  ( x –  ) 2 ( x –  ) 2 P(x) 1.167 2.167 3.167 4.167 5.167 6.167 -2.5 -1.5 -0.5 0.5 1.5 2.5 6.25 2.25 0.25 2.25 6.25 1.042.375.042.375 1.042 2.918

21. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 21 Standard Deviation of a single die roll: 1.707

22. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 22 Expected value and standard deviation… At a carnival, you decide to play spinner the wheel for a prize for $2.00. If you land in the red area, you win $4. If you land in the white area, you win $1. If you land in the blue area, you win nothing. What are your expected earnings when you play the game once? Ten times? Formula :  =  x P(x)

23. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 23 Standard Deviation

24. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 24 Finding the mean: xP(x) x P(x) red.333 white.333 blue.333 ???? ??? ??  =  x P(x) = ?

25. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 25 Finding the mean: xP(x) x P(x) $2.333 -$1.333 -$2.333 $0.67 -$0.33 -$0.67  =  x P(x) = -$0.33 If you do this ten times, you should expect to lose -$3.33

26. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 26 Finding the standard deviation xP(x) x –  ( x –  ) 2 ( x –  ) 2 P(x) $2.333 -$1.333 -$2.333 $2.33 –$0.67 –$1.67 $5.43 $0.45 $2.79 $1.81 $0.15 $0.93 $2.89

27. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 27 Standard Deviation $1.70

28. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 28 Linear Functions of a Random Variable If a and b are any constants and x is a random variable, then the new random variable L = a + bx is called a linear function of a random variable.

29. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 29 If x is a random variable with mean  and standard deviation , and L = a + bx then: Mean of L =  L = a + b  Variance of L =  L 2 = b 2  2 Standard deviation of L =  L = the square root of b 2  2 =  b  

30. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 30 If x is a random variable with mean = 12 and standard deviation = 3 and L = 2 + 5x Find the mean of L. Find the variance of L. Find the standard deviation of L.  L = 2 + 5    Variance of L = b 2  2 = 25(9) = 225 Standard deviation of L = square root of 225 = 

31. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 31 Independent Random Variables Two random variables x 1 and x 2 are independent if any event involving x 1 by itself is independent of any event involving x 2 by itself.

32. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 32 If x 1 and x 2 are a random variables with means    and   and variances     and     If W = ax 1 + bx 2 then: Mean of W =  W = a    + b   Variance of W =  W 2 = a 2  1 2 + b 2   2 Standard deviation of W =  W = the square root of a 2  1 2 + b 2   2

33. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 33 Given x 1, a random variable with  1 = 12 and  1 = 3 and x 2 is a random variable with  2 = 8 and  2 = 2 and W = 2x 1 + 5x 2. Find the mean of W. Find the variance of W. Find the standard deviation of W. Mean of W = 2(12)  + 5(8) = 64 Variance of W = 4(9) + 25(4) = 136 Standard deviation of W= square root of 136  11.66

34. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 34 Binomial Probability A special kind of discrete probability distribution with only 2 random variables

35. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 35 Features of a Binomial Experiment 1. There are a fixed number of trials. We denote this number by the letter n.

36. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 36 Features of a Binomial Experiment 2. The n trials are independent and repeated under identical conditions.

37. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 37 Features of a Binomial Experiment 3. Each trial has only two outcomes: success, denoted by S, and failure, denoted by F.

38. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 38 Features of a Binomial Experiment 4. For each individual trial, the probability of success is the same. We denote the probability of success by p and the probability of failure by q. Since each trial results in either success or failure, p + q = 1 and q = 1 – p.

39. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 39 Features of a Binomial Experiment 5. The central problem is to find the probability of r successes out of n trials.

40. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 40 Binomial Experiments Repeated, independent trials Number of trials = n Two outcomes per trial: success (S) and failure (F) Number of successes = r Probability of success = p Probability of failure = q = 1 – p

41. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 41 A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. Is this a binomial experiment?

42. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 42 Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. success = failure =

43. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 43 Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. success = hitting the target failure = not hitting the target

44. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 44 Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. Probability of success = Probability of failure =

45. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 45 Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. Probability of success = 0.70 Probability of failure = 1 – 0.70 = 0.30

46. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 46 Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. In this experiment there are n = _____ trials.

47. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 47 Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. In this experiment there are n = _8__ trials.

48. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 48 Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. We wish to compute the probability of six successes out of eight trials. In this case r = _____.

49. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 49 Is this a binomial experiment? A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. We wish to compute the probability of six successes out of eight trials. In this case r = _ 6__.

50. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 50 Binomial Probability as a Counting Method (FCP) = ? x ? x ? x ? x ? x ? x ? x ?  if taking 8 shots, we would multiply the probabilities together… = 0.7 x 0.7 x 0.7 x 0.7 x 0.7 x 0.7 x 0.3 x 0.3  if taking 8 shots, and 6 were successful and two were not… = 0.0106 … but that is only if the first 6 were successful and the last two were not… = 0.7 x 0.3 x 0.7 x 0.7 x 0.3 x 0.7 x 0.7 x 0.7  if taking 8 shots, and 6 were successful and two were not…in any particular order…how many different ways could it happen?

51. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 51 What are Binomial Experiments? The have a fixed number of _________ which we call the variable _____. The only outcomes for the binomial experiment are ___________ and ____________. Each trial in the experiment is ____________ and repeatable. Number of successes is the variable ____. Probability of success is the variable ____. Essentially each binomial experiment tries to the find the ____________ of __________ out of a given number of ____________.

52. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 52 Binomial Probability Formula

53. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 53 Calculating Binomial Probability Given n = 6, p = 0.1, find P(4):

54. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 54 Calculating Binomial Probability A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. n = 8, p = 0.7, find P(6):

55. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 55 Table for Binomial Probability Table 3 Appendix II

56. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 56 Using the Binomial Probability Table Find the section labeled with your value of n. Find the entry in the column headed with your value of p and row labeled with the r value of interest.

57. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 57 Using the Binomial Probability Table n = 8, p = 0.7, find P(6):

58. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 58 Find the Binomial Probability Suppose that the probability that a certain treatment cures a patient is 0.30. Twelve randomly selected patients are given the treatment. Find the probability that: a.exactly 4 are cured. b.all twelve are cured. c.none are cured. d.at least six are cured.

59. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 59 Exactly four are cured: n = r = p = q =

60. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 60 Exactly four are cured: n = 12 r = 4 p = 0.3 q = 0.7 P(4) = 0.231

61. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 61 All are cured: n = 12 r = 12 p = 0.3 q = 0.7 P(12) = 0.000

62. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 62 None are cured: n = 12 r = 0 p = 0.3 q = 0.7 P(0) = 0.014

63. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 63 At least six are cured: r = ?

64. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 64 At least six are cured: r = 6, 7, 8, 9, 10, 11, or 12 P(6) =.079 P(7) =.029 P(8) =.008 P(9) =.001 P(10) =.000 P(11) =.000 P(12) =.000

65. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 65 At least six are cured: P( 6, 7, 8, 9, 10, 11, or 12) =.079 +.029 +.008 +.001 +.000 +.000 +.000 =.117

66. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 66 Graph of a Binomial Distribution Binomial distribution for n = 4, p = 0.4:

67. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 67 Graph of a Binomial Distribution Binomial distribution for n = 4, p = 0.4: 0 1 2 3 4 r P( r ).4.3.2.1

68. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 68 Mean and Standard Deviation of a Binomial Distribution

69. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 69 Mean and standard deviation of the binomial distribution Find the mean and standard deviation of the probability distribution for tossing four coins and observing the number of heads appearing. n = 4 p = 0.5 q = 1 – p = 0.5

70. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 70 Mean and standard deviation of the binomial distribution

71. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 71 Find the mean and standard deviation: A sharpshooter takes eight shots at a target. She normally hits the target 70% of the time. Find the probability that she hits the target exactly six times. n = 8, p = 0.7

72. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 72 Geometric Distribution A probability distribution to determine the probability that success will occur on the nth trial of a binomial experiement

73. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 73 Geometric Distribution Repeated binomial trials Continue until first success Find probability that first success comes on nth trial Probability of success on each trial = p

74. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 74 Geometric Probability

75. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 75 A sharpshooter normally hits the target 70% of the time. Find the probability that her first hit is on the second shot. Find the mean and the standard deviation of this geometric distribution.

76. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 76 A sharpshooter normally hits the target 70% of the time. Find the probability that her first hit is on the second shot. P(2)=p(1-p) n-1 =.7(.3) 2-1 = 0.21 Find the mean  = 1/p = 1/.7  1.43 Find the standard deviation

77. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 77 Poisson Distribution A probability distribution where the number of trials gets larger and larger while the probability of success gets smaller and smaller

78. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 78 Poisson Distribution Two outcomes : success and failure Outcomes must be independent Compute probability of r occurrences in a given time, space, volume or other interval  (Greek letter lambda) represents mean number of successes over time, space, area

79. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 79 Poisson Distribution

80. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 80 The mean number of people arriving per hour at a shopping center is 18. Find the probability that the number of customers arriving in an hour is 20. r = 20 = 18Find P(20) e = 2.7183

81. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 81 The mean number of people arriving per hour at a shopping center is 18.

82. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 82 Poisson Probability Distribution Table Table 4 in Appendix II provides the probability of a specified value of r for selected values of.

83. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 83 Using the Poisson Table = 18, find P(20):

84. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 84 Poisson Approximation to the Binomial Distribution The Poisson distribution can be used as a probability distribution for “rare” events.

85. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 85 “Rare” Event The number of trials (n) is large and the probability of success (p) is small.

86. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 86 If n  100 and np < 10, then The distribution of r (the number of successes) has a binomial distribution which is approximated by a Poisson distribution. The mean = np.

87. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 87 Use the Poisson distribution to approximate the binomial distribution: n = 240 p = 0.02 Find the probability of at most 3 successes.

88. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 88 Using the Poisson to approximate the binomial distribution for n = 240 and p = 0.02 Note that n  100 and np = 4.8 < 10, so the Poisson distribution can be used to approximate the binomial distribution. Find the probability of at most 3 successes: Since = np = 4.8, we use Table 4 to find P( r  3) =.0082 +.0395 +. 0948 +. 1517 =.2942