[BruceZ]

The important thing to note is that even if your mean is not 0, or your standard deviation is not 1, you still use the standard normal distribution table to determine the probability for different numbers of standard deviations. All we are doing is changing units and shifting where you define 0. So if your mean is 100 bets and your standard deviation is 100 bets, then 100 bets corresponds to a standard deviation of 1 on the standard normal distribution, and 100 bets is the mean of the distribution. This would be your disribution after 100 hours if your mean is 1 bb/hr, and your standard deviation is 10 bets per hour (total mean = 1 bb/hr * 100 hours, total sigma = 10 bb/hr*sqrt(100 hours).

Also, I shouldn't say that we can do the "indefinite integral" from minus infinity to +infinity since this makes no sense. We cannot write the indefinite integral in terms of elementary functions, so we cannot do the definite integral between arbitrary limits. We are able evaluate the integral from minus infinity to +infinity.