Question

roulette is a casino game that involves players betting on where a ball will land on a spinning wheel. an american roulette wheel has 38 numbered slots - half of the slots from 1 to 36 are red and the other half are black. slots 0 and 00 are both green. suppose that a player bets $1 on the color red. if the ball lands in a red slot, the player gets their initial $1 back plus a payout of $1. if the ball doesnt land in a red slot, they lose their $1 bet. let x = the players net gain from a $1 bet on a the color red. here is the probability distribution of x:

calculate the mean of x. you may round your answer to the nearest thousandth.

e(x) = \square dollars

Explanation:

Step1: Recall the formula for the mean of a discrete random variable

The formula for the mean \(E(X)\) of a discrete random variable \(X\) is \(E(X)=\sum_{i}x_{i}P(x_{i})\), where \(x_{i}\) are the possible values of \(X\) and \(P(x_{i})\) are their corresponding probabilities.

Step2: Substitute the values into the formula

Here, \(x_1 = 1\) with \(P(x_1)=\frac{18}{38}\) and \(x_2=- 1\) with \(P(x_2)=\frac{20}{38}\).

$$
LATEXBLOCK0
$$

Step3: Simplify the fraction

\(\frac{-2}{38}=-\frac{1}{19}\approx - 0.053\)

Answer:

\(-0.053\) dollars