One of the wagers in the game of roulette is to place a bet that the ball will land on a red number. If the ball lands on a red number, the player wins the amount of his bet. If a player bets $5, find the player's expectation.
In a standard roulette wheel, there are 18 red numbers out of a total of 38 numbers (18 red, 18 black, and 2 green).
The probability of the ball landing on a red number is therefore 18/38, or 9/19.
If the player bets $5, the player will win $5 if the ball lands on a red number.
The expected value (or expectation) is calculated by multiplying the probability of winning by the amount won, and subtracting the probability of losing multiplied by the amount lost.
Expectation = (probability of winning)(amount won) - (probability of losing)(amount lost)
Expectation = (9/19)(5) - (10/19)(5)
Expectation = (45/19) - (50/19)
Expectation = -5/19
Therefore, the player's expectation is -$5/19. This means that on average, the player can expect to lose approximately $0.26 per $1 bet on a red number.
To find the player's expectation, we need to determine the probability of winning and losing, and the corresponding amounts.
In the game of roulette, there are 18 red numbers (out of 38 possible outcomes, including both red and black numbers).
Probability of winning (landing on a red number):
P(Winning) = Number of favorable outcomes / Total number of outcomes
= 18 / 38
≈ 0.4737
Probability of losing (landing on a black number):
P(Losing) = 1 - P(Winning)
= 1 - 0.4737
≈ 0.5263
The winning amount is the same as the bet, which is $5.
The expectation can be calculated as follows:
Expectation = (Probability of winning * Winning amount) - (Probability of losing * Losing amount)
= (0.4737 * $5) - (0.5263 * $5)
= $2.3685 - $2.6315
≈ -$0.263
Therefore, the player's expectation is approximately -$0.263, which means that on average, the player can expect to lose about $0.263 per bet.