Question

In American roulette, the wheel has the 38 numbers, 00, 0, 1, 2, ..., 34, 35, and 36, marked on equally spaced slots. If a player bets $5 on a number and wins, then the player keeps $5 and receives an additional $175. Otherwise, the player is awarded nothing, and the casino takes the player $5 e(x) to the player for one play of the game. If x is the gain to a player in a game of chance, then e(x) is usually negative. This value gives the average amount per game the player can expect to lose.

a) Expected value formula in roulette
b) Calculating average loss per game
c) Impact of winning and losing in roulette
d) Probability and outcomes in roulette

Answer 1

The expected value formula in roulette is calculated by multiplying each outcome by its probability and summing the products. The average loss per game can be calculated by subtracting the initial bet from the expected value.

Step-by-step explanation:

The expected value formula in roulette can be calculated by multiplying each outcome by its probability and summing the products. In American roulette, there are 38 slots on the wheel, including the numbers 00, 0, and 1 to 36. The probability of winning on a single number bet is 1/38, and if you win, you receive $175 in addition to your initial bet of $5. The expected value is calculated as (-$5) * (37/38) + ($170) * (1/38) = -$0.263, rounded to the nearest cent.

The average loss per game in roulette can be calculated by subtracting the initial bet from the expected value. In this case, the average loss per game is -$0.263 - (-$5) = -$4.737, rounded to the nearest cent.

Winning and losing in roulette have a significant impact on the expected value and average loss. Winning a single number bet results in a large payout of $175, while losing the bet results in a loss of $5. Since the probability of winning is low (1/38), the average loss per game is higher than the initial bet.

Probability and outcomes in roulette are determined by the number of slots on the wheel and the specific bets placed. Each number on the wheel has an equal probability of being hit, with a 1/38 chance. Different bets have different probabilities and payouts, affecting the overall outcome of the game.