Observe
Everyone has heard 20 buy-ins, or 30, or 50. The number by itself is useless, because the thing it protects against depends on two quantities nobody quotes alongside it. Here is the actual formula and what it says.
Treat results as the same random walk as before: drift w
big blinds an hour, standard deviation s big blinds an
hour. Starting from a bankroll of B big blinds, the
chance that the walk ever touches zero is
risk of ruin = e-2wB/s²
Three things fall straight out of that, and all three are worth more than the rule of thumb.
At a standard deviation of 50 big blinds an hour, with buy-ins of 100 big blinds:
| Win rate | 5 buy-ins | 10 buy-ins | 20 buy-ins | 30 buy-ins | 40 buy-ins |
|---|---|---|---|---|---|
| A small winner, 2.5 bb/hour | 36.8% | 13.5% | 1.83% | 0.248% | 0.0335% |
| A solid winner, 5 bb/hour | 13.5% | 1.83% | 0.0335% | under 0.001% | under 0.001% |
| A strong winner, 8 bb/hour | 4.08% | 0.166% | under 0.001% | under 0.001% | under 0.001% |
Chance of ever going broke, if the win rate and the swings stay what they are and you never move down. That last clause is doing a lot of work, and it is why these are floors rather than promises.
Look along the 20 buy-in column. For the solid winner it is 0.0335%, which is about as safe as anything in poker gets. For the small winner, the same 20 buy-ins is 1.83%, roughly 55 times riskier, at the same stake with the same swings.
The second bullet above is sitting in that table if you read it diagonally: 13.5% appears at 5 buy-ins for the solid winner and at 10 for the small one, and 1.83% appears at 10 and at 20. Doubling the win rate really does buy exactly what doubling the roll buys.
That is the part the rule of thumb hides. Twenty buy-ins is not a property of the game. It is a property of the game and how much you beat it for, and most people quoting it have never measured the second half.
w. A
player who withdraws their whole win rate has a risk of ruin of one.
The honest use of this table is not to find the smallest roll you can survive on. It is to see how much of your safety comes from the number you know least well, and to keep enough cushion that being wrong about it is survivable.
Both inputs to the formula come out of a session log and nowhere else. Log estimates your win rate and your standard deviation from your own completed sessions, in big blinds, so sessions at different stakes pool into one answer instead of three guesses.