I'm looking to calculate the standard deviation of a fixed-odds betting proposition. The bet pays 5/6 for a win and you lose your stake for a loss. For a bet of 100, a winner pays 83.33 (and the stake is returned) and a losing bet pays -100. The probability of winning is evens (or very close to evens).
I'm uncertain about the formula to use for Standard Deviation in this case. Is it
2* 1 [1 * sqrt(npq)]? In that case, the standard deviation of a single bet on the game above would be
100*sqrt[1 * 0.5*0.5]. Should it not be
2*100*sqrt[0.5*0.5] or something else, as a binomial distribution assumes a win of 1 and loss of 0 (or a win of 83.33 and a loss of 100 in this case), rather than a win of 1 and loss of -1 in this case.
I saw an example where a bet cost 20 and paid 100 if it was a winner, with the probability being 0.2. The standard deviation for that single bet was given as
100*sqrt[(0.2*0.8)]. I wanted to check whether this was right.
Thanks for any help on this.