Player Loss

One standard deviation = square root (N)
where N is the number of hands

99.7% of all results fall within three standard deviations of the mean.

Therefore, 99.85% of all results would fall to the right of minus three standard deviations.

If we were to solve for when 0 were -3 standard deviations from the mean we find

3 square root (N) = mean

mean = N x edge

3 square root (N) = 1.2% N

We now know that for one hand it is appropriate to rebate 1.186% of actual player loss, whereas at 62,500 hands, 50% of actual loss may be repaid with both scenarios maintaining a 50% equivalency relative to theoretical loss in an even money game.

To determine points in between these extremes of number of hands, it is necessary to determine the conditional mean for each number of hands. To crudely demonstrate the process of integration the following is provided:-

Number of hands N = 100
Edge = 1.2%
Even money game

13.64% of results occur between one and two standard deviations and

2.23% of results are greater than two standard deviations

From this we may roughly calculate the conditional mean for all player losses.

To do this we take the probability range and multiply this by the mid point result.

34.13% x {(1.2 + (1.2 + 10)) / 2 }
13.64% x {((1.2 + 10) + (1.2 + (2 x 10))) / 2 }
2.23% x {((1.2 + (2 x 10)) + (1.2 + (3 x 10))) / 2 }

and sum these which provides the conditional mean greater than the mean and then add the probability of results between 0 and the mean multiplied by that midpoint.

Without referring to normal distribution tables this may be approximated by taking the mean divided by the standard deviation and multiplying this by 34.13% and then multiplying that by the midpoint of zero and the mean.

= 1.2 / 10 x 34.13% x 1.2/2

Thus the conditional mean