One of the wagers in the game of roulette is to place a bet that the ball will land on a red number. (Eighteen of the numbers are black, 18 are red, and two are green.) 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.
The probability that the ball will land on a red number is $\dfrac{18}{38} = \dfrac{9}{19}$. The expectation is $\dfrac{9}{19} \cdot 5 = \boxed{\dfrac{45}{19}}$.
To find the player's expectation, we need to calculate the expected value of the bet.
The probability of the ball landing on a red number is 18/38 since there are 18 red numbers out of a total of 38 possible outcomes (18 red + 18 black + 2 green).
If the player wins the bet, they will receive $5. If they lose the bet, they will lose their $5.
Therefore, the player's expectation is calculated as follows:
Expected value = (Probability of winning * Amount won) + (Probability of losing * Amount lost)
Expected value = (18/38 * $5) + (20/38 * -$5)
Expected value = ($90/38) + (-$100/38)
Expected value = ($90 - $100)/38
Expected value = -$10/38
Hence, the player's expectation is expected to lose approximately -$0.26 per $5 bet.