Question

A warehouse contains ten printing machines, two of which are defective. A company selects seven of the machines at random, thinking all are in working condition. What is the probability that all seven machines are nondefective?

Answer 1

0.0667 = 6.67% probability that all seven machines are nondefective.

Explanation:

The machines are chosen from the sample 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:

10 machines means that
`n = 10`

2 defective, so 10 - 2 = 8 work correctly, which means that
`k = 8`

Seven are selected, which means that
`n = 7`

What is the probability that all seven machines are nondefective?

This is P(X = 7). So

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

`P(X = 7) = h(7,10,7,8) = (C_(8,7)*C_(2,0))/(C_(10,7)) = 0.0667`

0.0667 = 6.67% probability that all seven machines are nondefective.