Question

In the game of roulette, when a player gives the casino $20 for a bet on the number 12, the player has a 37/38 probability of losing $20 and a 1/38 probability of making a net gain of $700. (The prize is $720, but the player's $20 bet is not returned, so the net gain is $700.) If a player bets $20 that the outcome is an odd number, the probability of losing $20 is 20/38 and the probability of making a net gain of $20 is 18/38 .

a. If a player bets $20 on the number 12, what is the player's expected value?

Answer 1

The expected value when betting $20 on the number 12 in roulette is -$1.05. This means over time, a player would lose an average of $1.05 per game, indicating that it is not a profitable bet.

Step-by-step explanation:

Expected Value in Roulette

The expected value when a player bets $20 on the number 12 in roulette is calculated using the probabilities and the associated gains or losses for each outcome. The probability of losing the bet is 37/38, which would result in a loss of $20. Conversely, with a probability of 1/38, the player stands to make a net gain of $700. The formula to calculate the expected value (EV) is:

EV = (probability of losing) × (amount lost) + (probability of winning) × (net gain)

Using this formula, we calculate the expected value as follows:

EV = (37/38) × (-$20) + (1/38) × $700

EV = -$19.47 + $18.42

EV = -$1.05

Therefore, the player's expected value when betting $20 on the number 12 is -$1.05. If you play this game repeatedly, over a long string of games, you would expect to lose $1.05 per game, on average. Hence, the game is not favorable for a player aiming to win money over the long term as the expected value is negative.