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A container consists of twenty used electronic components of which five are defective.
Eight components are randomly taken from the container with replacement.
Determine the probability that at most two are defective.
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Twenty components; 5 are defective, 15 are good.
These are starting conditions, that are restored after every
of the 8 steps of the experiment (due to replacement).
So, the experiment is actually BINOMIAL with n= 8 trials and the probability
of "success" (getting defective component) p= = = 0.25.
The question is to find the probability of at most two success.
You may use the standard formula
P = P(n=8; k<=2; p=0.25) = .
Further, you may use free of charge online calculator
https://stattrek.com/online-calculator/binomial.aspx
and get the ANSWER in one click P = 0.679 (rounded).
Alternatively, you may calculate each term separately
= = 0.1,
= = 0.267,
= = 0.311,
and then calculate the sum 0.1 + 0.267 + 0.311 = 0.678, which is (almost) the same (modulo of rounding).
ANSWER. P = 0.679.
Solved.
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The major point in the solution is to get understanding that the experiment/question/problem
is about the Binomial distribution. As soon as you got it, the rest is just a technique.