Question

In order to win a prize, Heather randomly draws two balls from a basket of 40. There are 25 blue balls, and the rest are green balls. Of the blue balls, 12% are winning balls. Of the green balls, 20% are winning balls. Calculate the expected number of winning balls that Heather draws.

Answer 1

The expected number of winning balls that Heather draws is 0.3.

Explanation:

The balls are chosen 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)!)`

Expected value of the hypergeometric distribution:

The expected value is given by:

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

Expected number of blue and green balls:

40 balls, which means that
`N = 40`

2 are chosen, which means that
`n = 2`

25 are blue, which means that
`k = 25`

So

`E(X) = (nk)/(N) = (25(2))/(40) = 1.25`

1.25 balls are expected to be blue and 2 - 1.25 = 0.75 green.

Of the blue balls, 12% are winning.

Of the green balls, 20% are winning.

Calculate the expected number of winning balls that Heather draws.

`E_w = 1.25*0.12 + 0.75*0.2 = 0.3`

The expected number of winning balls that Heather draws is 0.3.