Question

one option in a roulette game is to bet $3 on red. (there are 18 red compartments, 18 black compartments, and two compartments that are neither red nor black.) if the ball lands on red, you get to keep the $3 you paid to play the game and you are awarded $3. if the ball lands elsewhere, you are awarded nothing and the $3 that you bet is collected. complete parts (a) through (b) below.
a. what is the expected value for playing roulette if you bet $3 on red?
$ (round to the nearest cent.)
b. what does this expected value mean? choose the correct statement below.
a. over the long run, the player can expect to break even.
b. this value represents the expected win over the long run for each game played.
c. this value represents the expected loss over the long run for each game played.

Explanation:

Step1: Calculate probability of winning

The total number of compartments is $18 + 18+2=38$. The probability of the ball landing on red, $P(\text{win})=\frac{18}{38}=\frac{9}{19}$. The probability of losing, $P(\text{lose})=\frac{18 + 2}{38}=\frac{20}{38}=\frac{10}{19}$.

Step2: Calculate the pay - offs

If you win, the net gain is $3$ (since you get back your $3$ and an additional $3$). If you lose, the net gain is $- 3$ (you lose the $3$ you bet).

Step3: Calculate the expected value

The formula for the expected value $E(X)$ of a discrete random variable is $E(X)=\sum_{i}x_ip_i$. Here, $x_1 = 3$ with probability $p_1=\frac{9}{19}$ and $x_2=-3$ with probability $p_2=\frac{10}{19}$. So $E(X)=3\times\frac{9}{19}+(-3)\times\frac{10}{19}=\frac{27}{19}-\frac{30}{19}=-\frac{3}{19}\approx - 0.16$.

Answer:

a. $-0.16$
b. C. This value represents the expected loss over the long run for each game played.