Final answer:
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.