On a roulette wheel, the probability of winning when
you pick a particular number is . Suppose you bet
$1.00 to play the game, and if your number is picked,
you get back $36.
a. Is this a fair game?
b. What would happen if you played this game a large
number of times?
It depends on the probability, which you have not posted.
To determine whether this is a fair game, we need to calculate the expected value. The expected value is the average amount of money you can expect to win or lose per bet.
a. To calculate the expected value, we need to multiply the probability of winning by the amount won, and subtract the probability of losing multiplied by the amount lost. In this case, the probability of winning is 1 in 38 (assuming there are 38 numbers on the roulette wheel, including 0 and 00), and the amount won is $36. The probability of losing is 37 in 38 (since there are 37 numbers other than the one you chose), and the amount lost is $1.
Expected value = (Probability of winning * Amount won) - (Probability of losing * Amount lost)
Expected value = (1/38 * $36) - (37/38 * $1)
Expected value = $0.947 - $0.974
Expected value = -$0.027
Since the expected value is negative, it means that on average, you would lose about $0.027 per bet. Therefore, the game is not fair, and the casino has an advantage.
b. If you played this game a large number of times, the law of large numbers states that the average outcome will approach the expected value. Since the expected value is negative, it means that over many bets, you would expect to lose money. However, in the short term, there can be considerable variability, and you could win or lose money while deviating from the expected value.