Question

There are 25 pens in a container on your desk. Among them, 20 will write well but 5 have defective ink cartridges. You will select 2 pens (without replacement) to take to a business appointment. Calculate the probability that both pens are defective.

Answer 1

3.33% probability that both pens are defective.

Explanation:

The pens are chosen without replacement, so we use the hypergeometric distribution to solve this question.

Hypergeometric distribution:

The probability of x sucesses 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 sucesses.

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:

2 defective, so x = 2.

25 pens, so N = 25.

Two pens will be selected, so n = 2.

5 are defective, so k = 5.

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

`P(X = 2) = h(2,25,2,5) = (C_(5,2)*C_(20,0))/(C_(25,2)) = 0.0333`

3.33% probability that both pens are defective.