The game of roulette uses a wheel containing 38 pockets. Thirty-six pockets are numbered 1, 2, 3, ..., 35, 36, and the remaining two are marked 0 and 00. The wheel is spun, and a pocket is identified as the "winner." Assume that the observance of any one pocket is just as likely as any other.
Suppose you bet $5 on a single number -- say, the number 18. The payoff on this type of bet is usually 35 to 1. What is your expected gain?
175(1/38) - 5(37/38) = ?
Of course, strictly speaking, you won't "win" exactly $175 on 18, because $5 of that was your original bet.
To calculate the expected gain in this situation, we need to consider the probability of winning and the potential payout.
In roulette, if you bet $5 on a single number and win, the payout is usually 35 to 1. This means that if you win, you will receive $35 plus your original $5 bet.
Since there are 38 pockets on the roulette wheel, the probability of winning with a single number bet is 1/38. This is because there is only one winning pocket out of the 38 total pockets.
To calculate the expected gain, we multiply the probability of winning by the potential payout and subtract the original bet:
Expected Gain = (Probability of Winning * Payout) - Bet
Expected Gain = (1/38 * $35) - $5
Expected Gain = ($35/38) - $5
Calculating the Expected Gain:
Expected Gain ≈ $0.921
Therefore, the expected gain from a $5 bet on a single number in roulette is approximately $0.92. This means that on average, for every $5 bet you make, you can expect to gain about $0.92. Keep in mind that this is an average over many bets, and individual results may vary.