Question

A game show airs on television five days per week. Each day, a prize is randomly placed behind one of two doors. The contestant wins the prize by selecting the correct door. What is the probability that exactly two of the five contestants win a prize during the week? Express your first answer as a fraction in simplest form, and round your percent answer to the nearest hundredth.

Answer 1

Probability = number of positive outcomes / number of total outcomes.

The number of total outcomes is 2 *2 *2 * 2* 2 = 32.

The positive outcomes are those where there are only two winners:

1)  w w _ _ _
2)  w _ w _ _
3)  w _ _ w _
4)  w _ _ _ w
5)  _ w w _ _
6)  _ w _ w _
7)  _ w _ _ w
8)  _ _ w w _
9)  _ _ w _ w
10) _ _ _ w w

Those are 10 outcomes where there are two w (winners).

Therefore, the probabilit is: 10 / 32 = 5 / 16

As percet that is: (5/16)*100 = 31.25%.

Answer: 5/16 or, equivalently, 31.25%

Answer 2

The probability that exactly 2 out of the 5 contestants win a price during the week is given by:

$P(X=2)={ ^5C_2\cdot\left(\frac{1}{2}\right)^2\cdot\left(\frac{1}{2}\right)^3 \\ \\ =\frac{10}{32}=\frac{5}{16}$

This is equivalent to 31.25%.