Question

Assuming a binomial distribution, a production process produces 2% defective parts. A sample of five parts from the production process is selected. What is the probability that the sample contains exactly two defective parts?

Answer 1

0.38% probability that the sample contains exactly two defective parts.

Explanation:

For each part, there are only two possible outcomes. Either it is defective, or it is not. The probabilities for each part being defective are independent from each other. So we use the binomial probability distribution to solve this problem.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

In this problem we have that:

`n = 5, p = 0.02`

What is the probability that the sample contains exactly two defective parts?

This is
`P(X = 2)`

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 2) = C_(5,2).(0.02)^(2).(0.98)^(3) = 0.0038`

0.38% probability that the sample contains exactly two defective parts.