[VideoReview]

Currently, to quickly determine the approximate probabilty of winning (or losing) any amount of money over any amount of bets (assuming I am always betting to win 1 unit), I use the following formula in Excel:

p = 1 - NORMSDIST((Total Won - Total Bet)/SQRT(Total Bet))

The above equation assumes that my edge is 0. Is there anyway to modify it for different edges? For example, if I have a system that I think might have a 5% edge, I might like to know the probability of the results in a different sample assuming this 5% edge and not an edge of zero.

[Ganchrow]

See this post on how to calculate the variance of a wager ik

In short it's:

σ²_k = (d_k - E_k - 1) * (E_k + 1) * B²_k

where σ²_k is the variance on the k^th bet,
d_k is the decimal odds on the k^th bet,
E_k is the percent edge of the k^th, and
B_k is the size of the k^th bet.

To calculate the total variance of N independent bets you'd just sum the individual variances of each separate bet.

σ²_TOTAL = Σ^N_i=1 { (d_i - E_i - 1) * (E_i + 1) * B²_i }

Taking the square root would then give you the standard deviation

Then to determine the p-value in Excel using the CLT you'd just take:

p-value = 1-NORMSDIST((Total Realized $ Return - Σ{B_i * E_i})/σ_TOTAL)

In other words, the CLT is allowing you to assume that the random variable (Realized Return - Expected Return) / (Standard Dev) is normally distributed with a mean of zero and standard deviation of 1. This assumption becomes increasingly more accurate as number of bets increases. With a smaller sample the t-distribution is probably a better approximation.