weighting population results

[gaming_mouse]

The avg of the sample means is still the maximum liklihood estimate of the population mean (i'm pretty sure).

Weighting the value here wouldn't really make sense. You are essentially saying that having more hands on the first player makes it more likely that the second player is like the first player -- which just isn't true.

The difference in hands does, however, effect the overall accuracy of you estimate of the population mean. Remember that the variance of the sum of two indepenent random variables is the sum of their variances. So it would come into play when calculating a C.I. for your estimate, eg.