The game of roulette uses a wheel containing 38 pockets. Thirty-six pockets are numbered 1,2.....,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?
prob of your number coming up is 1/36
expected gain = 35(5)(1/36) = 4.86
So, just like in every gambling game, in the long-run you would lose.
thank you
the Prob of your number coming up is 1/38 , not 1/36
which changes the result to
5(35)(1/38) = 4.61
even worse than before
To calculate the expected gain of betting $5 on a single number in the game of roulette, we need to consider the probability of winning and the potential payoff.
In this scenario, there are 38 possible pockets on the wheel. Since you are betting on a single number (e.g., 18), there is only 1 winning pocket out of the 38. Therefore, the probability of winning is 1/38.
The payoff for this type of bet is usually 35 to 1. This means that if you win, you will receive 35 times your original bet, plus get back your original bet. So, if you bet $5, you would receive (35 * $5) + $5 = $180.
To calculate the expected gain, multiply the probability of winning by the payoff and subtract the original bet:
Expected gain = (Probability of winning) * (Payoff) - (Original bet)
Expected gain = (1/38) * $180 - $5
Expected gain = $4.74 - $5
Expected gain = -$0.26
Therefore, the expected gain from betting $5 on a single number in this scenario is -$0.26, meaning you can expect to lose an average of $0.26 on each bet in the long run.