What is risk of ruin in poker and how do you calculate it

What risk of ruin measures

Wikipedia defines risk of ruin as the likelihood of losing all of one's capital, or of the bankroll falling below the minimum needed to keep playing. Its simplest example is betting everything on one coin toss: the risk of ruin is 50 percent. Poker is a long sequence of much smaller bets, so the question becomes whether the bankroll ever touches zero somewhere along the way, before a winning edge carries it far enough from zero to be safe.

PrimeDope, which publishes poker variance calculators, describes it as a function of three numbers: your win rate, your standard deviation, and your bankroll. The win rate is how fast money comes in on average. The standard deviation is how far individual results scatter around that average. The bankroll is the money set aside for poker and nothing else. A single bad night does not ruin anyone; risk of ruin is about the whole path of sessions, including the bad runs that every winning player eventually hits.

The formula and where it comes from

The version most poker players use is RoR = e^(−2 × WR × BR ÷ SD²). In PrimeDope's form, WR is the win rate in big blinds per 100 hands, BR is the bankroll in big blinds, SD is the standard deviation in big blinds per 100 hands, and e is Euler's number, about 2.718. The only rule is consistency: if the win rate is per 100 hands, the standard deviation must be per 100 hands too.

The formula treats the bankroll as a random walk that drifts upward at the win rate while being pushed around by variance, the same model mathematicians call Brownian motion with drift. A lecture from MIT's Advanced Stochastic Processes course (15.070J, lecture 7) shows that the highest point reached by Brownian motion with a negative drift follows an exponential distribution. Turned upside down, that means the chance of a winning bankroll ever falling a given distance below its start shrinks exponentially with the distance. That is why the bankroll sits inside the exponent.

Bill Chen and Jerrod Ankenman cover the topic at length in The Mathematics of Poker. According to the Library of Congress table of contents, chapter 22 is titled "Staying in Action: Risk of Ruin", chapter 23 "Adding Uncertainty: Risk of Ruin with Uncertain Win Rates", and chapter 24 deals with growing bankrolls and the Kelly criterion. The second of those chapters exists because the win rate you plug in is itself only an estimate.

Worked examples

PrimeDope gives three. A micro-stakes grinder with a 7 bb/100 win rate, a 75 bb/100 standard deviation and a 2,000 big blind bankroll has a risk of ruin of about 0.7 percent. A mid-stakes regular at 3 bb/100, 85 bb/100 and 2,500 big blinds is at about 12.5 percent. A marginal winner at 1.5 bb/100, 90 bb/100 and 2,000 big blinds is at about 47.7 percent, close to a coin flip.

Because the bankroll is in the exponent, doubling it squares the risk. The mid-stakes regular with 5,000 big blinds instead of 2,500 drops from 12.5 percent to about 1.6 percent. Halving the win rate does the opposite and takes the square root: at 1.5 bb/100 with the same 2,500 big blinds the risk rises to about 35 percent. Swings matter twice because the standard deviation is squared, so 10 percent more spread, 93.5 instead of 85, lifts 12.5 percent to about 18 percent.

Live players can do the same arithmetic in dollars and hours, since the time unit cancels out as long as both the win rate and the standard deviation use it. PokerCharts estimates a full-ring no-limit standard deviation of roughly 80 to 100 bb/100 and a live pace of 20 to 30 hands an hour. At 90 bb/100 and 25 hands an hour, that is about 45 big blinds an hour, or $225 an hour at $2/$5. A player winning $15 an hour with a $5,000 bankroll then has a risk of ruin of about 5.2 percent.

Moving up to $5/$10 with the same $5,000 and the same skill measured in big blinds doubles both the dollar win rate and the dollar standard deviation. The exponent halves, and the risk becomes the square root of the old figure: about 22.7 percent. Playing twice as big on the same money is mathematically the same as cutting the bankroll in half.

Getting your own numbers

1. Win rate: total profit divided by total hours played, or by hands if you have hand histories.
2. Standard deviation: for each session, compare the result with what your average rate predicts for that many hours, square the difference, divide by the hours, average those values across sessions and take the square root. That gives a standard deviation per hour that can be compared across sessions of different lengths.
3. Bankroll: only money you can lose without touching rent, savings or anything else. If you would top it up from your paycheck after a bad month, the formula's closed-system assumption no longer holds, and the result describes a different situation from yours.

Common mistakes with risk of ruin

Trusting a short-sample win rate. A few dozen sessions can make a breakeven player look like a strong winner, and the formula will then print a comfortingly small number. PokerCharts describes the first 100 hours of live results as almost entirely noise. This is the problem Chen and Ankenman's chapter on uncertain win rates is about.

Forgetting the assumptions. PrimeDope lists them: a constant win rate, normally distributed results (real poker results have somewhat fatter tails), no moving down in stakes, and no deposits or withdrawals. Taking money out for living costs raises the real risk; moving down after losses lowers it. Recalculate whenever any of the inputs changes.

Plugging in a losing or breakeven win rate. The formula only makes sense for a positive win rate. With no edge, or a negative one, a random walk eventually reaches any level below its start, so over unlimited play the risk is 100 percent whatever the bankroll size. A bigger bankroll only buys time.

Moving up on the same bankroll without redoing the arithmetic, as the $5/$10 example shows. RollGauge applies this same formula to the sessions you log, labels the reading an early estimate until ten sessions are in, and shows 100 percent with a rough count of remaining sessions when your pace is breakeven or losing.

This guide explains bankroll arithmetic. It is not financial advice and does not suggest that poker is a reliable way to make money. If gambling is causing problems for you or someone close to you, the National Council on Problem Gambling's helpline, 1-800-MY-RESET (1-800-697-3738), is confidential and open 24/7 by call, text or chat.

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