Question

A shipment of 50 precision parts including 4 that are defective is sent to an assembly plant. The quality control division selects 10 at random for testing and rejects the entire shipment if 1 or more are found defective. What is the probability this shipment passes inspection?

Answer 1

0.3968 = 39.68% probability this shipment passes inspection.

Explanation:

The parts are chosen without replacement, which means that the hypergeometric distribution is used to solve this question.

Hypergeometric distribution:

The probability of x successes is given by the following formula:

`P(X = x) = h(x,N,n,k) = (C_(k,x)*C_(N-k,n-x))/(C_(N,n))`

In which:

x is the number of successes.

N is the size of the population.

n is the size of the sample.

k is the total number of desired outcomes.

Combinations formula:

`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)!)`

In this question:

50 parts means that
`N = 50`

4 defective means that
`k = 4`

10 are chosen, which means that
`n = 10`

What is the probability this shipment passes inspection?

Probability that none is defective, so:

`P(X = x) = h(x,N,n,k) = (C_(k,x)*C_(N-k,n-x))/(C_(N,n))`

`P(X = 0) = h(0,50,10,4) = (C_(4,0)*C_(46,10))/(C_(50,10)) = 0.3968`

0.3968 = 39.68% probability this shipment passes inspection.