Question

A roulette wheel consists of [18] red slots, [18] black slots, and [2] green slots. A player can wager [1] that a marble will land in a red slot, in which case the player would gain a [1]. If the marble lands in a slot that isn't red, then the player loses their [1] wager. Let [x] represent the player's gain from a random [1] wager on red. What is the probability distribution of [x] along with the summary statistics?

Answer 1

The probability distribution for a player's gain, x, on a red wager in roulette is Prob(x=1) = 18/38 and Prob(x=-1) = 20/38. Summary statistics include the expected value, variance, and standard deviation of this distribution.

Step-by-step explanation:

The probability distribution of a player's gain, x, from a random wager on red in a roulette game can be determined by looking at the possible outcomes and their probabilities. Since there are 18 red slots, the probability of winning (gain of 1) is 18/38, because there are 18 red slots out of a total of 38 slots. If the ball lands on a non-red slot (black or green), which has a combined total of 20 slots, the player loses their wager (gain of -1), so the probability of losing is 20/38.

The probability distribution of x is as follows:

• Prob(x=1) = 18/38
• Prob(x=-1) = 20/38

The summary statistics for this distribution include the expected value, variance, and standard deviation.
The expected value (E(x)) is calculated as: E(x) = (1)(18/38) + (-1)(20/38).

The variance (Var(x)) is calculated using the formula Var(x) = E(x^2) - [E(x)]^2, where E(x^2) is the expected value of x squared.

The standard deviation is the square root of the variance.