Question

one option in a roulette game is to bet $6 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 $6 you paid to play the game and you are awarded $6. if the ball lands elsewhere, you are awarded nothing and the $6 that you bet is collected. complete parts (a) through (b) below.
a. what is the expected value for playing roulette if you bet $6 on red?
$ (round to the nearest cent.)
b. what does this expected value mean? choose the correct statement below.
a. this value represents the expected win over the long run for each game played.
b. this value represents the expected loss over the long run for each game played.
c. over the long run, the player can expect to break even.

Explanation:

Step1: Calculate probability of winning

Total compartments = 18 + 18+ 2=38. Probability of ball landing on red $P(\text{win})=\frac{18}{38}$.

Step2: Calculate probability of losing

Probability of ball not landing on red $P(\text{lose}) = 1-\frac{18}{38}=\frac{20}{38}$.

Step3: Determine winning and losing amounts

If win, net gain is $6$ (since you get back your $6$ bet and an additional $6$). If lose, net gain is $- 6$ (you lose your $6$ bet).

Step4: Calculate expected - value

Expected value $E(X)=6\times\frac{18}{38}+(-6)\times\frac{20}{38}=\frac{108 - 120}{38}=\frac{- 12}{38}\approx - 0.32$.

Answer:

a. -$0.32$
b. B. This value represents the expected loss over the long run for each game played.