Question

If 3 balls are drawn at random from a bag containing 3 red and 4 blue balls,

what is the expected number of red balls in the sample?

Answer 1

The expected number of red balls in the sample is 1.2857.

Explanation:

The balls are chosen without replacement from the sample, 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)!)`

The expected value of the hypergeometric distribution is:

`E(X) = (nk)/(N)`

3 red balls in the sample:

This means that
`k = 3`

3 balls are drawn:

This means that
`n = 3`

Total of 3 + 4 = 7 balls:

This means that
`N = 7`

What is the expected number of red balls in the sample?

`E(X) = (nk)/(N) = (3*3)/(7) = 1.2857`

The expected number of red balls in the sample is 1.2857.