Question

one option in a roulette game is to bet $28 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 $28 you paid to play the game and you are awarded $28. if the ball lands elsewhere, you are awarded nothing and the $28 that you bet is collected. find the expected payback for this roulette game if you bet $28 on red. the expected payback is $□. (round to the nearest cent as needed.)

Explanation:

Step1: Calculate the probability of winning

The total number of compartments is \(18 + 18+2=38\). The number of red compartments (winning cases) is \(n = 18\). So the probability of winning \(P(\text{win})=\frac{18}{38}\).

Step2: Calculate the probability of losing

The probability of losing \(P(\text{lose})=\frac{38 - 18}{38}=\frac{20}{38}\).

Step3: Calculate the pay - off for winning and losing

If you win, the pay - off \(X_1=\$28\) (you get back your bet of \(\$28\) and win \(\$28\)). If you lose, the pay - off \(X_2=-\$28\) (you lose the \(\$28\) you bet).

Step4: Use the expected value formula

The expected value formula is \(E(X)=\sum_{i}X_iP(X_i)\). So \(E(X)=X_1P(\text{win})+X_2P(\text{lose})\).
Substitute the values: \(E(X)=28\times\frac{18}{38}+(- 28)\times\frac{20}{38}\).
Factor out \(28\): \(E(X)=28\times(\frac{18 - 20}{38})\).
Simplify: \(E(X)=28\times(-\frac{2}{38})=-\frac{56}{38}\approx - 1.47\)

Answer:

\(-1.47\)