Here's something I did a while ago that may be interesting. I was investigating how long is the long term in poker, given some modest input assumptions.



The graphs show a 95% CI on the projected BR for a winrate of 5 BB/100 and variance of 80 BB/100.

The dotted green line is the EV, or mean value. The red and cyan lines are the upper and lower bounds of the CI.

You can see in the lower plot that the short term is dominated by variance, with the boundaries expanding proportionally to the square root of the number of hands played.

In the long term plot above, you can see that the winrate (no matter how slight) will eventually dominate. The EV increases linearly, while the boundaries increase proportional to the square root. The linear function will always dominate in the long run.

Note that the upper graph is a logarithmic scale. The green line is the same straight line in the lower plot, but the "squashing" of the x-axis makes it look curved. The red and cyan lines are always an equal distance above and below the green line. The oddness in the logarithmic graph is an artifact of the 'squishing.'

The logarithmic scale is chosen to illustrate just how long the long term is in poker.