Question

Your neighbor has a bag with 5 oranges and 7 apples in it. You will be receiving two pieces of fruit from your neighbor. What is the probability, in percent, that you will receive 2 apples, assuming she removes them from the bag in random order

Answer 1

31.82% probability that you will receive 2 apples.

Explanation:

The fruits are removed from the bag, which means that the hypergeometric distribution is used 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:

5 + 7 = 12 total fruits, which means that
`N = 12`

7 apples, which means that
`k = 7`

You receive 2 fruits, which means that
`n = 2`

What is the probability, in percent, that you will receive 2 apples, assuming she removes them from the bag in random order?

This is, as a proportion, P(X = 2). So

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

`P(X = 2) = h(2,12,2,7) = (C_(7,2)*C_(5,0))/(C_(12,2)) = 0.3182`

0.3182*100% = 31.82%

31.82% probability that you will receive 2 apples.