Question

When playing roulette at a​ casino, a gambler is trying to decide whether to bet $5 on the number 25 or to bet $5 that the outcome is any one of the three possibilities, 00, 0, or 1. The gambler knows that the expected value of the $5 bet for a single number is -53 cents. For the $5 bet that the outcome is 00, 0, or 1, there is a probability of 3/38 of making a net profit of $20 and a 35/38 probability of losing $5.

Find the expected value for the ​$5 bet that the outcome is 00, 0, or 1.

Answer 1

The expected value of a $5 bet that the roulette outcome is 00, 0, or 1 is -$3.03, suggesting an average loss of $3.03 per game.

Step-by-step explanation:

The expected value for a bet in roulette decides whether a gamble is worth taking based on the average amount a player can expect to win or lose per bet if they made the same bet many times over. In the case of betting $5 that the outcome is one of three possibilities (00, 0, or 1), we must calculate the expected value using the probabilities and outcomes given.

The calculation of the expected value is as follows:

• There is a 3/38 chance to make a net profit of $20, and
• a 35/38 chance of losing $5.

So, the expected value (EV) is calculated by multiplying each outcome by its respective probability and summing these products:

EV = (3/38 * $20) + (35/38 * (-$5))

EV = (3/38 * 20) - (35/38 * 5)

EV = (3 * 0.5263) - (35 * 0.1316)

EV = 1.5789 - 4.6053

EV = -$3.0263

Therefore, the expected value of making the $5 bet on the outcomes 00, 0, or 1 is approximately -$3.03, indicating an average loss of $3.03 per game when making this bet.