Binominal Distributions Worksheet With Answer Key - Helm 2008 Section 372 Page 7
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Key Point 4
The Binomial Probabilities
Let X be a discrete random variable, being the number of successes occurring in n independent
trials of an experiment. If X is to be described by the binomial model, the probability of exactly r
successes in n trials is given by
P(X = r) = C p q
.
Here there are r successes (each with probability p), n
r failures (each with probability q) and
n!
C =
is the number of ways of placing the r successes among the n trials.
r!(n
r)!
Notation
If a random variable X follows a binomial distribution in which an experiment is repeated n times
each with probability p of success then we write X
B(n, p).
Example 9
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 4 compo-
nents, find the probabilities that X takes the values 0 to 4.
Solution
From Example 8, we know that the probabilities required are the terms of the expansion of the
expression:
4
(0.9 + 0.1)
so
X
B(4, 0.1)
Hence the required probabilities are (using the general formula with n = 4 and p = 0.1)
4
P(X = 0) = (0.9)
= 0.6561
3
P(X = 1) = 4(0.9)
(0.1) = 0.2916
4
3
2
2
P(X = 2) =
(0.9)
(0.1)
= 0.0486
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2
4
3
2
3
P(X = 3) =
(0.9)(0.1)
= 0.0036
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2
3
4
P(X = 4) = (0.1)
= 0.0001
4
Also, since we are using the expansion of (0.9 + 0.1)
, the probabilities should sum to 1, This is a
useful check on your arithmetic when you are using a binomial distribution.
23
HELM (2008):
Section 37.2: The Binomial Distribution
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