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October 27th, 2017, 07:33 AM   #1
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Binomial Distribution Probability

A toy manufacturing company tests the quality of the toys it manufactures. On a daily basis, 20 toys are taken at random for testing and to check that 95% meet the toy specification. The day's production will be accepted provided that no more than 2 toys fail to meet the specification standard.

Calculate the probability if the inspectors pass the day's toy production as acceptable if 75% of the day's toys meet the specification.

n=20
Success(P)=0.75
Failure(Q)=0.25

=(^20 C 19)(0.75)^19(0.25)^20−19

=0.0211

The result seems lower than what I would expect. Where have I gone wrong?
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October 27th, 2017, 08:27 AM   #2
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"No more than 2" does not mean "1". Instead, it means "0, 1 and 2".
Therefore, it should be
(^20 C18 )(0.75)^18(0.25)^20−18 +
(^20 C19 )(0.75)^19(0.25)^20−19 +
(^20 C20 )(0.75)^20(0.25)^20−20 =
0.06694781+0.02114141+0.003171212=0.09126043


To make it easier, use this free calculator online by inputting (2/20/0.25), and you will automatically get the cumulative probability of 0.09126043 for no more than 2 failures.
Here, as the target is to calculate failure, we use 0.25 instead of 0.75.
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