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I've seen quoted in a number of places that the standard deviation for a 1-number bet on a 38 number Roulette wheel (0, 00, 1, 2, .., 36) is 5.76. I can't seem to find a single source that shows how this calculation is made.

I am a student, this is for further understanding of a homework problem, so please assume I'm clueless and be as detailed as possible.

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What are your possible outcomes if you bet $1 on a single number, meaning how much do you win if that number comes up and how much do you lose if that number does not come up? What are the probabilities of these events? – jsk Jun 10 '14 at 3:03
I didn't notice the mention of homework before. Since it's in connection to a homework problem, please add the self-study tag, and check the tag wiki info. When you say 'further understanding' I assume you're not actually doing a homework problem, just trying to understand one... is that right? – Glen_b Jun 10 '14 at 3:11
One of the first hits on a Google search is the Wikipedia article which shows the calculation and relates it to the theory of the Binomial distribution. – whuber Jun 10 '14 at 5:45
up vote 7 down vote accepted

Your question should make explicit what quantity you want the standard deviation of.

You say "a one number bet" but you don't clarify what the outcome is that you're considering.

I will assume you mean the following:

Let there be a bet of one unit (say \$1), with a payout of 35-1 -- that is you stake \$1 and you end up with an outcome of either \$0 or \$36 (equivalently, a profit of either \$-1 or \$35). The probability of the outcome \$36 is 1/38. What is the standard deviation of the outcome?

Let's start first with the variance. (The variance is the square of the standard deviation.)

See the definition of variance here, and specifically for a discrete random variable here.

The outcome $0$ has probability 37/38 and the outcome $36$ has probability 1/38.

So the variance is $E(X^2)-E(X)^2 = \frac{36^2\times 37}{38}-(\frac{36\times 37}{38})^2 = 33.20776...\quad$

and the standard deviation is the square root of that.

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PERFECT! Thanks so much, this makes a lot of sense. – Johnny Bones Jun 10 '14 at 3:18

sorry, I don't have enough reputation for a comment, this is a comment to Glen_b's answer. When $X$ is the outcome, isn't $E(X) = 0\cdot 37/38 + 36 \cdot 1/38$ ?

It leads to the same variance though.

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So it's boiled down to (36^2/38) - (36/38)^2? – Johnny Bones Jun 10 '14 at 12:56
exactly. It gives 33.207 like @Glen_b said. – spore234 Jun 10 '14 at 14:22

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