Risk of ruin: the formula, worked numbers and what it means

Risk of ruin (often shortened to RoR) answers one question: if I keep making this bet, at this size, how likely am I to lose everything I set aside for it?

"Everything" means the bankroll you have dedicated to the activity, not your life savings, although for some gamblers those end up being the same thing. "Ruin" is reaching zero, or reaching a level where you can no longer place the next bet.

Three inputs drive the answer.

1. Edge per bet. Positive, zero or negative expected value per unit staked.
2. Variance per bet. How widely each result can swing. A 14x long shot swings much more than an even-money bet with the same edge.
3. Bankroll in units. Not dollars, but how many bets of your chosen size the bankroll covers. $1,000 at $10 a bet is 100 units; the same $1,000 at $100 a bet is 10 units.

A fourth input, the goal, turns the question into a race: will you hit the target before you hit zero? Without a goal, the question becomes whether you ever go broke over an unlimited number of bets.

This page covers the mathematics. To size a real session budget with your own numbers, use the bankroll calculator. For blackjack-specific ruin figures from simulations, the blackjack simulator guide keeps those.

The classic model has you start with i units and bet 1 unit at a time on an even-money bet. You win each bet with probability p and lose with probability q = 1 − p. You stop at 0 (ruin) or at N (target).

The problem is old. The earliest known mention is in a 1656 letter from Blaise Pascal to Pierre de Fermat, and Christiaan Huygens included a version in his 1657 treatise on games of chance, one of the first printed books on probability.

Fair game (p = q = 1/2)

P(ruin) = 1 − i / N

Start with 10 units and aim for 20, and your risk of ruin is 50%. Aim for 11 and it is 1 − 10/11 ≈ 9.1%. Aim for 100 and it is 90%. On a fair game, ruin probability depends only on how far the target is compared with your bankroll.

Unfair game (p ≠ q)

Let r = q / p. Then:

P(ruin) = (rⁱ − rᴺ) / (1 − rᴺ)

No target, playing forever

• If p ≤ q (fair or negative edge), ruin is certain: P(ruin) = 1.
• If p > q (positive edge), P(ruin) = (q / p)ⁱ.

The fair case being certain ruin surprises people. A fair random walk eventually visits every level, including zero. The only escape is a finite goal or finite time.

Beyond even money

For bets with other payouts, a widely used approximation is RoR ≈ exp(−2 × μ × B / σ²), where μ is the edge per unit bet, σ² is the variance per unit bet and B is the bankroll in units. It only applies when μ is positive. For an even-money bet with a 2% edge it gives almost exactly the same answers as the exact formula.

Take an even-money bet that wins with probability 7/15 ≈ 46.67%, such as Purple on a 15-slot wheel paying 2x. The edge is −1/15 ≈ −6.67%, and r = q/p = (8/15)/(7/15) = 8/7.

Trying to double a bankroll with 1-unit bets

On a fair coin the chance of doubling never changes. On a negative-edge bet it collapses as the number of steps grows, because more steps give the edge more time to work. Taking 50 small steps towards a goal means 50 units of negative drift to fight through.

Small targets with a big bankroll

Start with 100 units and aim to finish 10 units ahead. On a fair coin the risk of ruin is 1 − 100/110 ≈ 9.1%. At 7/15 it is about 73.7%. Being 90 units away from ruin and only 10 from the goal is not enough protection when every step drifts the wrong way.

Chance of ever being ahead

With unlimited bankroll on the 7/15 bet, the chance of ever reaching +k units is (p/q)ᵏ = (7/8)ᵏ.

No bankroll, however large, improves on those numbers. They are a ceiling set by the edge alone.

A positive edge changes the question from "when" to "whether". With p > q and no target, the risk of ever going broke is (q/p) raised to the bankroll in units.

Suppose a bettor has a genuine 2% edge on an even-money proposition: p = 0.51, q = 0.49.

Two patterns stand out.

• Even a real edge can go broke. With 20 units, a 2% edge still loses everything nearly half the time. Variance does not care that you are right on average.
• Doubling the bankroll squares the risk. 13.5% at 50 units becomes 0.135² ≈ 1.8% at 100 units, because (q/p)¹⁰⁰ = ((q/p)⁵⁰)². The same effect comes from halving your bet size.

Higher-variance bets need more units

The approximation exp(−2μB/σ²) shows why variance matters as much as edge. Take a hypothetical game with a 1% edge and a standard deviation of 1.15 units per bet, a figure in the range often quoted for blackjack hands. With 100 units, RoR ≈ exp(−2 × 0.01 × 100 / 1.3225) ≈ 22%. With 300 units it drops to about 1%.

Proportional betting avoids ruin in theory

If you bet a fixed fraction of your current bankroll instead of a fixed amount, you can never technically reach zero, because each bet shrinks as the bankroll shrinks. That is the idea behind the Kelly criterion. In practice minimum bet sizes, and the pain of a bankroll shrinking towards nothing, bring back a practical version of ruin.

If the edge is against you and you need to reach a specific target, the ruin formula points to an unexpected conclusion: bet big.

Say you have 50 units and must reach 100. Betting 1 unit at a time on the 7/15 bet gives a 0.13% chance of success. Betting all 50 units once gives 46.67%. Betting 25 units at a time wins twice in a row with probability (7/15)² ≈ 21.8%; a win and a loss returns you to 50 units to try again, and counting those repeats the overall chance is p² / (1 − 2pq) ≈ 43.4%. That is slightly below all-in but hundreds of times better than timid play.

Lester Dubins and Leonard Savage formalised this in their 1965 book *How to Gamble If You Must*: in a sub-fair casino with a fixed goal, bold play (staking as much as needed to reach the goal, or everything you have if less) maximises the probability of reaching it.

This is not a recommendation. It is a statement about a narrow problem: one fixed target, a negative edge, and nothing else you care about. Most people gamble for entertainment over time, which is the opposite problem, and for that goal small bets make the budget last. The lesson for everyone is the same, though. Small bets on a negative edge do not reduce the cost. They spread it over more rounds. The expected loss is always the edge times the total amount wagered, as house edge explains.

Progressions and ruin

Doubling systems such as the Martingale strategy look like they lower risk because most sessions end with a small win. What they actually do is concentrate risk into rare, large losses that reach ruin quickly. A constant stake, covered in flat betting, makes the risk easier to see and to budget.

The story of Archie Karas, who reportedly ran a small stake into a fortune and then lost it, is often retold as a risk-of-ruin lesson: at a negative edge, a bankroll that keeps playing keeps being exposed.

PVPspinArena runs three player-vs-player games in USDC or ETH on Base, and the ruin maths maps onto each one.

• [Roulette](/roulette): Purple and Silver are the 7/15 bet used in the tables above. Green pays 14x on 1 slot in 15, with the same 6.67% edge but far higher variance, so a bankroll of Green bets reaches ruin with bigger swings along the way.
• [Coinflip](/coinflip): two players on a 50/50. With no fee it is the fair-coin column, where risk of ruin is 1 − i/N. Any fee shown before entry pulls the numbers towards the negative-edge column.
• [Jackpot](/): your win chance equals your share of the pot. Small shares are long shots, which behave like high-variance bets: long losing runs are normal.

Every result comes from committed seeds and can be checked on fairness. Fair draws are what make the formulas on this page reliable.

The practical move is to decide your ruin level before you start: the amount you are prepared to lose in a session, measured in bets. If that number is small, your risk of hitting it is high, and that is fine as long as it is money set aside for entertainment. Deposit limits and breaks are on responsible gambling. Play is 18+. More systems and odds guides sit in the games and odds topic.

In the same cluster, see also labouchere system.