Binominal Distributions Worksheet With Answer Key - Helm 2008 Section 372 Page 6
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20Note that the probabilities you have obtained:
3
2
2
3
q
, 3q
p, 3qp
, p
3
3
2
2
3
are the terms which arise in the binomial expansion of (q + p)
= q
+ 3q
p + 3qp
+ p
Task
Repeat the previous Task for the binomial model for the case with n = 4.
Your solution
Answer
Number of successes
4
3
2
1
0
4
3
2
2
3
4
Probability
p
4p
q 6p
q
4pq
q
Again we explore the connection between the probabilities and the terms in the binomial expansion
4
of (q + p)
. Consider this expansion
4
4
3
2
2
3
4
(q + p)
= q
+ 4q
p + 6q
p
+ 4qp
+ p
3
Then, for example, the term 4p
q, is the probability of 3 successes in the four trials. These successes
3
can occur anywhere in the four trials and there must be one failure hence the p
and q components
which are multiplied together. The remaining part of this term, 4, is the number of ways of selecting
three objects from 4.
4!
4
Similarly there are
C
=
= 6 ways of selecting two objects from 4 so that the coefficient 6
2
2!2!
2
2
combines with p
and q
to give the probability of two successes (and hence two failures) in four
trials.
The approach described here can be extended for any number n of trials.
22
HELM (2008):
Workbook 37: Discrete Probability Distributions
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