Poker Risk of Ruin Calculator

This poker risk of ruin calculator estimates the probability that a cash-game bankroll eventually reaches zero under a fixed-stakes model. Enter winrate, standard deviation and bankroll, or use the inverse modes to find a required bankroll or winrate for a chosen risk target.

The fixed-stakes exponential approximation is RoR = exp(-2μB/σ²): μ is expected winrate, B is bankroll and σ² is variance. For a winrate of 5 bb/100, SD 90 and a bankroll of 2,000 bb, it gives 8.47% eventual risk of ruin. The same bankroll at SD 100 gives 13.53%.

How to use the poker risk of ruin calculator

Choose the mode that answers your question:

• Risk of Ruin: given your bankroll, winrate and SD, what is the modeled probability of ever going broke?
• Required Bankroll: given winrate, SD and a target probability, how many big blinds or complete buy-ins meet that target?
• Required Win Rate: given bankroll, SD and a target, how large must the expected edge be within the model?

Use big blinds for bankroll and bb/100 for winrate and SD. One buy-in on this page is 100 bb. Enter 1 for a 1% risk target in the tool. After editing an input or choosing a game preset, press Calculate to refresh the result. Changing modes recalculates using the fields relevant to the new question.

The required-winrate output is a mathematical threshold, not evidence that you can achieve that edge. For dollar conversions and comparisons with the 20-buy-in rule, use the bankroll requirements tool.

How the Formula Works

The model represents bankroll movement as a steady expected gain plus random fluctuations. With a positive winrate, the eventual probability of reaching zero is:

r = exp(−2 × w × B / s²)

Here w and s are the same quantities represented by μ and σ in the traditional notation. Solving that expression in the other direction gives:

• Required bankroll: B = −s² × ln(r) / (2 × w).
• Required winrate: w = −s² × ln(r) / (2 × B).

The probability r is a fraction: 0.01 means 1%. The logarithm is the natural logarithm. The tool converts its percentage input before calculating and squares SD once; entering variance in the SD field would square it a second time.

Variable Definitions

Time is measured in blocks of 100 hands. The tracker convention “SD bb/100” describes the spread in big blinds over a block of 100 hands; its variance is measured in bb². There is no extra division by 100 in the ruin formula. Divide the bankroll by 100 only to express it in 100 bb buy-ins.

The positive-edge exponential formula assumes w > 0, s > 0 and B > 0. With zero or negative winrate and positive SD, eventual ruin is 100% for every finite bankroll in this model. The direct mode reports that result; the bankroll mode reports that no finite amount meets a target below 100%.

Shared standard-deviation presets

These are the same illustrative game scenarios used by our cash-game variance and bankroll tools. They are not measured population averages or a way to infer your SD from a style label. Use data from a comparable game, stake and stack depth where available.

The default NLHE 6-max preset is 90. The SD 100 tables below remain separate, explicitly labeled reference scenarios. A different SD is a different input, even when the underlying formula is the same.

Spreadsheet and Code Formulas

For the compact spreadsheet expressions below, use A1 for winrate, B1 for bankroll, C1 for SD and D1 for target probability. B1 and C1 must be positive; inverse calculations require D1 strictly between 0 and 1. Set D1 to 1% or 0.01 for a 1% target, not the number 1.

For a positive winrate, the basic Excel or Google Sheets expression is:

=EXP(-2*A1*B1/(C1^2))

Its result is a fraction. Format the cell as Percentage, or multiply by 100 and display a plain number; do not do both. To also handle a zero or negative winrate under the model:

=IF(A1<=0,1,EXP(-2*A1*B1/(C1^2)))

For positive A1 and C1, a bankroll rounded up to the next big blind is:

=ROUNDUP(-C1^2*LN(D1)/(2*A1),0)

To round the requirement up to whole 100 bb buy-ins instead:

=ROUNDUP(-C1^2*LN(D1)/(200*A1),0)

The required winrate, rounded upward to four decimal places, is:

=ROUNDUP(-C1^2*LN(D1)/(2*B1),4)

These formulas assume valid numeric cells and the stated domains. For example, A1 = 5, B1 = 2000 and C1 = 100 produces approximately 0.135335, or 13.53%. With D1 = 1%, the bankroll formula returns 4,606 bb and the whole-buy-in formula returns 47.

Python risk-of-ruin function

import math

def poker_ror(winrate, bankroll, stddev):
    values = (winrate, bankroll, stddev)
    if not all(math.isfinite(x) for x in values):
        raise ValueError("Inputs must be finite numbers")
    if bankroll <= 0 or stddev <= 0:
        raise ValueError("Bankroll and SD must be positive")
    if winrate <= 0:
        return 1.0
    return math.exp(-2 * winrate * bankroll / stddev**2)

Example: Why Variance Hurts Winning Players

Consider a net winrate of 5 bb/100 with SD 100 and 2,000 bb of starting capital:

r = exp(−2 × 5 × 2000 / 100²) = exp(−2) ≈ 0.135335

That is 13.53% eventual risk under the model. For a 1% target, the unrounded bankroll threshold is 4,605.1702… bb. Rounding upward gives 4,606 bb, or 47 whole buy-ins = 4,700 bb.

These are different bankroll amounts with different rounding units. Forty-six buy-ins contain only 4,600 bb and give approximately 1.0052% risk, just above the 1% target. The difference is small, but a minimum requirement should not be rounded below its threshold.

Required winrate example

Keeping the bankroll at 2,000 bb and SD at 100, a 1% target requires an unrounded winrate of 11.512925… bb/100. The tool rounds that threshold up to 11.5130 bb/100. This reverses the equation; it does not establish that such an edge is available in your game.

Cross-check at SD 90

With SD 90 and an expected edge of 5 bb/100, a 1% target instead gives 3,730.1879… bb, rounded up to 3,731 bb or 38 whole buy-ins. The 5% and 2% targets give 25 and 32 BI. These agree with the updated bankroll and variance pages for the same inputs.

Bankroll Tables (Quick Reference)

The first two tables calculate risk for a specified starting bankroll. Their buy-in column is the actual bankroll divided by 100, so fractional units are retained. The third table solves for a target and rounds required bb and whole buy-ins upward independently.

Table 1: Win Rate = 5 bb/100, Standard Deviation = 100 bb/100

Table 2: Win Rate = 3 bb/100, Standard Deviation = 100 bb/100

This table changes winrate while holding SD at 100. It is a sensitivity comparison, not a recommendation for a particular level of stakes.

Table 3: Bankroll Required for 1% Estimated Bust Probability

Within each cell, the bb figure and whole-buy-in figure are separate rounded alternatives, not equal amounts. For example, at 5 bb/100 and SD 100, 4,606 bb meets the model target, while 47 complete buy-ins contain 4,700 bb.

Mathematical Background

Let T count blocks of 100 hands and let W(T) be standard Brownian motion. Before absorption at zero, the continuous model is X(T) = B + wT + sW(T). The wT term is expected gain and sW(T) supplies normally distributed fluctuations with variance s²T.

The event of interest is whether this process ever reaches zero, starting from B. For positive drift, the lower-barrier hitting probability is exp(−2wB/s²). Karl Sigman’s Brownian-motion notes, §1.8 give the equivalent upper-barrier result for negative drift; reflecting the process gives the ruin formula used here.

This result is exact for the continuous model. Its application to real poker is an approximation. A poker hand has discrete outcomes, and winrate, SD and playing conditions can change. The model alone does not establish the direction or size of its error for an actual game.

Reading the bankroll simulation

The chart starts with the same 5/90/2,000 scenario as the main tool, over 100,000 hands, but has independent controls. Its analytical percentage describes eventual ruin. The sample fraction counts paths that reach zero within the chosen hand count. A small sample can differ substantially from either its finite-horizon probability or the eventual figure.

Each 100-hand increment has mean w and standard deviation s, using a normal draw. To account for crossings between positive checkpoints x and y, the simulator uses the conditional probability exp(−2xy/(s²ΔT)), where ΔT is measured in 100-hand blocks. The Brownian-bridge treatment in Mike Giles’s Lecture 10 gives this barrier-crossing result.

Paths stop permanently at zero. The chart places the final point at the end of the interval in which ruin occurred; it does not estimate the exact crossing time. The green line B + wT is the mean of the process without stopping at zero, not the average of the stopped paths. Reset and a fixed seed make the illustrated sample reproducible.

Key Assumptions

1. Fixed stakes: bankroll is measured in units of a constant big blind.
2. Constant edge and SD: the inputs describe the assumed underlying game, not a changing sample estimate.
3. Independent increments: the model has no tilt, game selection or serial dependence.
4. No external cash flows: the bankroll is neither replenished nor reduced by withdrawals.
5. Infinite horizon: the analytical percentage concerns ever reaching zero, with no upper limit on hands.
6. Positive edge for finite inverse bankrolls: with non-positive winrate and positive SD, every finite bankroll eventually reaches zero with probability one in this model.

What This Model Does Not Capture

• A fixed session length: eventual ruin is not the probability of losing money after 10,000 or 100,000 hands.
• Maximum drawdown: a drop from a prior peak need not reach zero. It requires a separate path-based calculation and a horizon.
• Uncertain winrate: an observed winning sample does not prove a positive long-term edge. Compare several plausible inputs.
• Rake and rewards changing over time: use net winrate, avoid subtracting rake twice, and treat uncertain or delayed rewards separately.
• Move-down rules, deposits and withdrawals: these alter bankroll evolution. Their effect depends on the policy and on how the game changes.
• Tournament payouts: cash-game bb/100 inputs are not tournament ROI or a substitute for a payout-distribution model.

Lowering the risk target raises the required bankroll. It does not resolve uncertainty about your edge or guarantee that you can afford a loss. There is no universal percentage here that makes play “safe” or suitable for a professional.

References and Further Reading

Poker-specific

Bill Chen and Jerrod Ankenman, The Mathematics of Poker, ConJelCo, 2006, provides further reading on quantitative poker methods.

Mathematical Background

Karl Sigman, Notes on Brownian Motion, §1.8, Proposition 1.4, is the mathematical reference for the hitting probability. The inverse formulas in this article follow by taking logarithms and rearranging.

Online Resources

The NIST normal-distribution reference explains the Gaussian distribution used for Brownian increments and outcome ranges. Its role differs from the barrier-hitting result: a final-result distribution alone does not give the probability of crossing zero along the way.

Frequently Asked Questions