Final answer:
The expected value for the casino over 800 games of this probability-based game is $37,776 to the nearest dollar.
Step-by-step explanation:
The expected value for the casino in this game is calculated by considering the probability of the casino winning and the payouts for winning and losing. Since the casino has a 98% chance of winning, the expected value for the casino per game can be found by multiplying the probabilities and payouts accordingly. The casino gains $59 when they win and loses $530 (the $59 stake plus the $471 winnings) when they lose.
Calculation of Expected Value
The probability of the casino losing is therefore 2% or 0.02. The expected value (EV) for the casino per game is:
EV = (probability of winning × gain from a win) + (probability of losing × loss from a loss)
EV = (0.98 × $59) + (0.02 × -$530)
EV = $57.82 - $10.60
EV = $47.22
Thus, the expected value for the casino for each game is $47.22. To find the expected value over 800 games, we multiply this amount by 800:
Total EV for 800 games = $47.22 × 800 = $37,776
To the nearest dollar, the expected value for the casino over 800 games would be $37,776.