To determine how long it will take for your average win to be 1 standard deviation, divide the square of your standard deviation for 1 hour by the square of your hourly rate.

hours to be ahead 84% of the time = (sigma/ev)^2.
*84% = the 68% of the time you will be +/- 1SD and the 16% you will be up more than 1SD.

This is one way to define the "long run". To find out how long it will take for your hourly rate to equal let's say 1.6 standard deviations, this is simply

hours to at least break even 95% of the time = (1.6*sigma/ev)^2.

At this point you have a 95% probability of being ahead.


Lets say we have a limit player with a 2BB/100 "true" win rate, and a 20BB/100 standard deviation.

How long will it take for his actual win to be GREATER THAN ZERO 99% of the time?


(2.3*20/2)^2 = 529*100 = 52,900 hands

So there is still a 1% chance that he will be down money at the end of FIFTY THREE THOUSAND hands.
All of us who have played for some time have had runs >2000 hands where we were down a SIGNIFICANT amount.