Calculating SD

How does one calculate the standard deviation on a bet with uneven payoffs given the probability of those payoffs? For example,

50-1 with probability 0.01,
25-1 with probability 0.01,
10-1 with probability 0.02,
Lose with probability 0.96.

The expectation is 0.01*50 + 0.01*25 + 0.02*10 + 0.96*(-1) = -0.01. What is the standard deviation and how do you derive it?

Take each member of the population, square the difference between it and the mean of the population, and multiply by its probability. The sum of these values is the variance. The square root of the variance is the standard deviation.

50-1 with probability 0.01,
25-1 with probability 0.01,
10-1 with probability 0.02,
Lose with probability 0.96.

The expectation is 0.01*50 + 0.01*25 + 0.02*10 + 0.96*(-1) = -0.01.

You have 4 different results. The mean result is -.01.

[50 - .01]^2 * .01+
[25 - .01]^2 * .01+
[10 - .01]^2 * .02+
[-1 - .01]^2 * .96=
Variance

Again, the sqrt of the variance is the SD. I don't think that you will have a normal distribution here. This means the mean may not be the median, and one SD to the right may not be the 84 percentile.

Thanks. That's clear now.

I don't think that you will have a normal distribution here. This means the mean may not be the median, and one SD to the right may not be the 84 percentile.

Good point. Would the following formula for risk of ruin still apply (assuming positive EV)?

R = ((1 - EV/SD)/(1 + EV/SD))^(Bank/SD)

(I hope I remembered that correctly.)

Where is it in your formula???

Using commonsense, with negative ev, ruin is guaranteed as number of trials tends towards infinity.

Mathematically, Lim--->& R=1 ev<0

I think you missed out something regarding no of wagers.

Some give you chance of ruin before hitting a target win, some give chance of ruin before hitting a target number of bets, and some give you chance of ruin for infinite bets. Equations generally assume that all wins go back into the bank until the criterion, number of bets or target win, is attained. For infinite N ruin equations it is assumed that the bank builds forever and no money is ever removed. It essentially gives the probability of ruin before you aquire all the wealth in the universe or play till the end of time.