Variance calculator
Poker results swing far more than most players expect, and the swing has nothing to do with how well you play. Put in your win rate and see how wide the range is over a number of hands, how often a winning player is still losing, and the bankroll that keeps the risk of going broke small.
How the figures are worked out
The expected result is the win rate times the number of hands divided by 100. The spread grows with the square root of the number of hands: the standard deviation of the whole sample is the per-100 figure times the square root of hands divided by 100. Over 20,000 hands that is 80 times the square root of 200, about 1,130 big blinds, against an expected win of 1,000. The two ranges are one and 1.96 standard deviations either side of the expectation, which is where about 68% and 95% of results fall if they follow the normal curve, as poker results do closely enough over thousands of hands.
The chance of being down is the chance that a result at least that far below the expectation turns up. With a win rate of 5 and a standard deviation of 80 it is about 19% after 20,000 hands and about 27% after 10,000. A winning player who is losing after a month has not necessarily done anything wrong.
The risk of ruin uses the usual approximation for a player with a steady positive win rate: the risk with a bankroll of B big blinds is e to the power of minus two times the win rate times B divided by the standard deviation squared. Turned round, the bankroll for a given risk is the standard deviation squared, divided by twice the win rate, times the natural log of one over the risk. For the bankroll calculator this is the reason behind 30 to 50 buy-ins: at 100 big blinds a buy-in, 1,900 big blinds is 19 buy-ins for a 5% risk, and the rule adds a margin for a win rate that is smaller than you think.
Questions
What standard deviation should I use?
For six-handed no-limit Hold'em cash games, 80 to 100 big blinds per 100 hands is typical. Full-ring tables are lower, around 60 to 80. Pot-limit Omaha is higher, often 120 or more. A tracking program shows your own figure after a few thousand hands.
What is a good win rate?
At small stakes online, 2 to 5 big blinds per 100 hands is solid, and 10 is excellent and rare. Rakeback adds to it: 30% rakeback on 7 big blinds of rake per 100 hands is about 2 big blinds per 100 more.
How many hands before I know whether I am winning?
More than feels right. With a win rate of 5 and a standard deviation of 80, the 95% range after 20,000 hands is still from about a loss of 1,200 big blinds to a win of 3,200, so a small win rate cannot be told from zero in a month of play. Over 100,000 hands the picture is much clearer.
What is risk of ruin?
The chance of losing the whole bankroll before the win rate carries you clear. For a player with a positive win rate it falls quickly as the bankroll grows: with a win rate of 5 and a standard deviation of 80, about 1,900 big blinds gives a 5% risk and about 2,900 gives 1%. A player with a win rate of zero or less goes broke eventually whatever the bankroll.