Binominal Distributions Worksheet With Answer Key - Helm 2008 Section 372 Page 4
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20Example 8
A worn machine is known to produce 10% defective components. If the random
variable X is the number of defective components produced in a run of 3 compo-
nents, find the probabilities that X takes the values 0 to 3.
Solution
Assuming that the production of components is independent and that the probability p = 0.1 of
producing a defective component remains constant, the following table summarizes the production
run. We let G represent a good component and let D represent a defective component.
Note that since we are only dealing with two possible outcomes, we can say that the probability q of
the machine producing a good component is 1
0.1 = 0.9. More generally, we know that q+p = 1
if we are dealing with a binomial distribution.
Outcome Value of X
Probability of Occurrence
3
GGG
0
(0.9)(0.9)(0.9) = (0.9)
2
GGD
1
(0.9)(0.9)(0.1) = (0.9)
(0.1)
2
GDG
1
(0.9)(0.1)(0.9) = (0.9)
(0.1)
2
DGG
1
(0.1)(0.9)(0.9) = (0.9)
(0.1)
2
DDG
2
(0.1)(0.1)(0.9) = (0.9)(0.1)
2
DGD
2
(0.1)(0.9)(0.1) = (0.9)(0.1)
2
GDD
2
(0.9)(0.1)(0.1) = (0.9)(0.1)
3
DDD
3
(0.1)(0.1)(0.1) = (0.1)
From this table it is easy to see that
3
P(X = 0) = (0.9)
2
P(X = 1) = 3
(0.9)
(0.1)
2
P(X = 2) = 3
(0.9)(0.1)
3
P(X = 3) = (0.1)
Clearly, a pattern is developing. In fact you may have already realized that the probabilities we have
3
found are just the terms of the expansion of the expression (0.9 + 0.1)
since
3
3
2
2
3
(0.9 + 0.1)
= (0.9)
+ 3
(0.9)
(0.1) + 3
(0.9)(0.1)
+ (0.1)
We now develop the binomial distribution from a more general perspective. If you find the theory
getting a bit heavy simply refer back to this example to help clarify the situation.
First we shall find it convenient to denote the probability of failure on a trial, which is 1
p, by q,
that is:
q = 1
p.
What we shall do is to calculate probabilities of the number of ‘successes’ occurring in n trials,
beginning with n = 1.
n n n = = = 1 1 1
With only one trial we can observe either 1 success (with probability p) or 0 successes
(with probability q).
20
HELM (2008):
Workbook 37: Discrete Probability Distributions
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