Question

Naval intelligence reports that 99 enemy vessels in a fleet of 1818 are carrying nuclear weapons. If 99 vessels are randomly targeted and destroyed, what is the probability that no more than 11 vessel transporting nuclear weapons was destroyed

Answer 1

0.001687 = 0.1687% probability that no more than 1 vessel transporting nuclear weapons was destroyed.

Explanation:

The vessels are destroyed and then not replaced, 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:

Fleet of 18 means that
`N = 18`

9 are carrying nuclear weapons, which means that
`k = 9`

9 are destroyed, which means that
`n = 9`

What is the probability that no more than 1 vessel transporting nuclear weapons was destroyed?

This is:

`P(X \leq 1) = P(X = 0) + P(X = 1)`

In which

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

`P(X = 0) = h(0,18,9,9) = (C_(9,0)*C_(9,9))/(C_(18,9)) = 0.000021`

`P(X = 1) = h(1,18,9,9) = (C_(9,1)*C_(9,8))/(C_(18,9)) = 0.001666`

Then

`P(X \leq 1) = P(X = 0) + P(X = 1) = 0.000021 + 0.001666 = 0.001687`

0.001687 = 0.1687% probability that no more than 1 vessel transporting nuclear weapons was destroyed.