With knowledge of your standard deviation (SD) and mean (win rate, EV, whatever you want to call it), you can determine your risk of ruin (ROR) for a given bankroll (or vice versa). Also, you can do things such as assess how likely a losing or winning streak was.
Sources
Risk of ruin equation (Thank you BruceZ)
Computing your standard deviation (Thank you Mason)
Examples
Let's say you have an EV of 1 BB/hr and an SD of 10 BB/hr (which you solved for with the assistance of Mason's article). Here are some things you can do:
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1)</font> You have a bankroll of 300 BB's, and would like to know what your ROR is:
r = exp(-2uB/sigma^2) = exp(-2*1*300/10^2) = exp(-6) =
~.25%
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2)</font> You would like to have a risk of ruin of 1%, and are wondering what size bankroll is required:
B = -(sigma^2/2u)ln(r) = -(10^2/2*1)*ln(.01) =
~230 BB's
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3)</font> You have just suffered through a 150 BB losing streak over your last 50 hours of play. You would like to know how often you should expect to lose that much or more:
a) Solve for your EV over a 50 hr stretch:
EV/hr = 1 BB
EV/50 hrs = 50*1 = 50 BB's
b) Solve for your SD over a 50 hr stretch:
SD/hr = 10 BB
SD/50 hrs = 10*sqrt(50) = 70.7 BB's
c) Determine how many negative standard deviations you have suffered through to lose 150 BB's:
50 + 70.7*x = -150
70.7*x = -200
x = -2.83 SD's
d) Look up N(-2.83) in a probability table or use Excel (go into a cell and type <font color="green">=normsdist(-2.83)</font>):
The value is
.234%, or
1/427.
-- Homer