1. Chapter Five The Binomial Probability Distribution and Related Topics
Understandable Statistics Seventh Edition By Brase and Brase Prepared by: Lynn Smith Gloucester County College
Chapter Five
The Binomial Probability Distribution and Related Topics

2. Statistical Experiment
A statistical experiment or observation is any process by which an measurements are obtained

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

4. a quantitative variable that assumes a value determined by chance
Random Variable
a quantitative variable that assumes a value determined by chance

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. 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. Probability Distribution
an assignment of probabilities to the specific values of the random variable or to a range of values of the random variable

8. 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.

9. Probability distribution for the rolling of an ordinary die
x P(x) 1 2 3 4 5 6

10. Features of a Probability Distribution
x P(x) 1 2 3 4 5 6
Probabilities must be between zero and one (inclusive)
 P(x) =1

11. Probability Histogram
P(x)
| | | | | | |

12. Mean and standard deviation of a discrete probability distribution
Mean =  = expectation or expected value,
the long-run average
Formula:  =  x P(x)

13. Standard Deviation

14. Finding the mean:  =  x P(x) = 1.4 0 .3 1 .3 .3 2 .2 .4 3 .1 4 .1
x P(x) x P(x)
0 .3
1 .3
2 .2
3 .1
4 .1 .3 .4
 =  x P(x) = 1.4
1.4

15. Finding the standard deviation
x P(x) x –  ( x – ) ( x – ) 2 P(x)
0 .3
1 .3
2 .2
3 .1
4 .1
.588
.048
.072
.256
.676
– 1.4
– 0.4 .6
1.6
2.6
1.96
0.16
0.36
2.56
6.76
1.64

16. Standard Deviation
1.28

17. 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.

18. 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 = b2  2
Standard deviation of L =  L= the square root of b2  2 = b 

19. 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 = b2  2 = 25(9) = 225
Standard deviation of L = square root of 225 = 

20. Independent Random Variables
Two random variables x1 and x2 are independent if any event involving x1 by itself is independent of any event involving x2 by itself.

21. If x1 and x2 are a random variables with means and and variances and  If W = ax1 + bx2 then:
Mean of W =  W = a + b 
Variance of W = W 2 = a2  12 + b2  2
Standard deviation of W =  W= the square root of a2  12 + b2  2

22. Given x1, a random variable with 1 = 12 and  1 = 3 and x2 is a random variable with  2 = 8 and  2 = 2 and W = 2x1 + 5x2.
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

23. Binomial Probability

24. Features of a Binomial Experiment
1. There are a fixed number of trials.
We denote this number by the letter n.

25. Features of a Binomial Experiment
2. The n trials are independent and repeated under identical conditions.

26. Features of a Binomial Experiment
3. Each trial has only two outcomes:
success, denoted by S,
and failure, denoted by F.

27. 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.

28. Features of a Binomial Experiment
5. The central problem is to find the probability of r successes out of n trials.

29. 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

30. A sharpshooter takes eight shots at a target
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?

31. 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 =

32. 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

33. 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 =

34. 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

35. 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.

36. 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.

37. 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 = _____.

38. 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__.

39. Binomial Probability Formula

40. Calculating Binomial Probability
Given n = 6, p = 0.1, find P(4):

41. 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):

42. Table for Binomial Probability
Appendix II

43. 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.

44. Using the Binomial Probability Table
n = 8, p = 0.7, find P(6):

45. Find the Binomial Probability
Suppose that the probability that a certain treatment cures a patient is 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.

46. Exactly four are cured:
n =
r =
p =
q =

47. Exactly four are cured:
n = 12
r = 4
p = 0.3
q = 0.7
P(4) = 0.231

48. All are cured:
n = 12
r = 12
p = 0.3
q = 0.7
P(12) = 0.000

49. None are cured:
n = 12
r = 0
p = 0.3
q = 0.7
P(0) = 0.014

50. At least six are cured:
r = ?

51. At least six are cured: r = 6, 7, 8, 9, 10, 11, or 12 P(6) = .079

52. At least six are cured: P( 6, 7, 8, 9, 10, 11, or 12)=
= .117

53. Graph of a Binomial Distribution
Binomial distribution for n = 4, p = 0.4:

54. Graph of a Binomial Distribution
Binomial distribution for n = 4, p = 0.4:
P( r ) .4 .3 .2 .1 r

55. Mean and Standard Deviation of a Binomial Distribution

56. 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

57. Mean and standard deviation of the binomial distribution

58. 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

59. Geometric Distribution
A probability distribution to determine the probability that success will occur on the nth trial of a binomial experiement

60. 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

61. Geometric Probability

62. 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.

63. 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

64. Poisson Distribution
A probability distribution where the number of trials gets larger and larger while the probability of success gets smaller and smaller

65. 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

66. Poisson Distribution

67. 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
 = 18 Find P(20)
e =

68. The mean number of people arriving per hour at a shopping center is 18.

69. Poisson Probability Distribution Table
Table 4 in Appendix II provides the probability of a specified value of r for selected values of .

70. Using the Poisson Table
 = 18, find P(20):

71. Poisson Approximation to the Binomial Distribution
The Poisson distribution can be used as a probability distribution for “rare” events.

72. “Rare” Event
The number of trials (n) is large and the probability of success (p) is small.

73. 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.

74. Use the Poisson distribution to approximate the binomial distribution:
Find the probability of at most 3 successes.

75. 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) = = .2942