What is the standard deviation of a $1 bet on any one particular number, e.g., 33, in the game of
American roulette? Notice the payoff is 35:1 for this type of bet.
To calculate the standard deviation of a $1 bet on a particular number in American roulette, we need to understand the concept of standard deviation and the probabilities associated with the game.
In American roulette, there are 38 possible outcomes since there are 36 numbers (1-36) plus a 0 and a 00. The probability of winning on a single number bet is 1/38, because there is only one winning outcome out of the total 38 possible outcomes.
The payoff for this type of bet is 35:1, meaning if you win, you receive 35 times your original bet plus your original bet back. For a $1 bet, if you win, you would receive $35 + $1 = $36.
To calculate the standard deviation of this bet, we need to consider the range of possible outcomes and their associated probabilities. We will assume that losing the bet results in a negative return of the original bet ($1), while winning the bet gives a positive return of $35.
We can create a table to calculate the probabilities and returns:
Outcome | Probability (P) | Return (X)
-----------------------------------------
Lose | 37/38 | -$1
Win | 1/38 | $35
Now, we can calculate the expected return (mean) of this bet:
Expected Return (mean) = (Probability of losing * Return when losing) + (Probability of winning * Return when winning)
= (37/38 * (-$1)) + (1/38 * $35)
= (-$37/38) + ($35/38)
≈ -$0.0526
As we can see, the expected return is slightly negative, indicating that, on average, a $1 bet on a single number in American roulette would lead to a loss in the long run.
To calculate the standard deviation, we need to square the difference between each return and the mean, multiply it by the probability, and then take the square root of the sum of these values:
Standard Deviation = Square root of [(Probability of losing * (Return when losing - Expected Return))^2 + (Probability of winning * (Return when winning - Expected Return))^2]
Standard Deviation = Square root of [(37/38 * (-$1 - (-$0.0526))^2 + (1/38 * ($35 - (-$0.0526)))^2]
Standard Deviation ≈ $4.551
Therefore, the standard deviation of a $1 bet on any one particular number in American roulette is approximately $4.551.