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

Answer:

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

Step-by-step 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:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}$

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.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

Expected value of the hypergeometric distribution:

The expected value is given by:

$E(X) = \frac{nk}{N}$

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

$E(X) = \frac{nk}{N} = \frac{25(2)}{40} = 1.25$

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.