Risk of Ruin in Gambling: The Formula With Examples

Risk of ruin is the probability that a bankroll hits zero before it reaches a chosen goal. For repeated even-money bets of one unit each, the answer comes from a short formula. In a fair game, the chance of reaching a goal of N units from a bankroll of k units is k ÷ N, so doubling a 100-unit bankroll succeeds 50% of the time. With a negative edge, the same goal becomes far less likely: on an American roulette wheel, betting red, doubling 100 units before ruin has a probability of only about 0.0027%.

The key point is that ruin risk depends on three things: the size of the bankroll measured in bets, how far away the goal is, and the edge on each bet. A negative edge pulls every path toward zero, and the more bets are placed, the stronger that pull becomes. If play continues without a goal, ruin is certain in the long run for any negative-edge game.

This guide gives the gambler’s ruin formulas with each symbol defined, works the roulette example step by step, compares bankroll sizes in a table, and shows how a positive-edge case would differ. Results are probabilities for a stated model, not forecasts.

What Risk of Ruin Measures

Imagine a simple model. You hold a bankroll of k units and place one-unit even-money bets. Each bet wins one unit with probability p and loses one unit with probability q = 1 − p. You stop when you reach a goal of N units or when you have nothing left. Reaching zero is called ruin, and the probability of reaching it first is the risk of ruin.

This model treats the bankroll as a random walk with two absorbing barriers, 0 and N. Because the steps are random, variance decides how quickly the walk touches a barrier, and the edge decides which barrier it is more likely to touch. The variance in gambling guide explains the first part; the house edge supplies the second.

Risk of ruin is not the same as losing a session. A session can end below its starting point without the bankroll ever reaching zero. It is also not a measure of how much is lost on average, which is the expected loss, and it is not a prediction. It is the probability of one specific event, under one specific set of assumptions.

Units matter. A $1,000 bankroll with $10 bets is 100 units; the same bankroll with $50 bets is only 20 units. Measuring the bankroll in bets, not dollars, is what makes the formulas comparable across stakes.

The Gambler’s Ruin Formulas

Let P(k) be the probability of reaching the goal N from a bankroll of k units. Each bet moves the bankroll up or down by one unit, so P(k) = p × P(k + 1) + q × P(k − 1), with P(0) = 0 and P(N) = 1. Solving that relationship gives two cases.

where:

k = starting bankroll in units.
N = goal in units (the bankroll at which play stops).
p = probability of winning one even-money bet, q = 1 − p.
r = q ÷ p, the ratio of losing to winning probability. It is above 1 for a negative-edge game.
rᵏ = r multiplied by itself k times.

For an American wheel, betting red, p = 18 ÷ 38 ≈ 0.4737 and q = 20 ÷ 38, so r = 20 ÷ 18 ≈ 1.1111. On a European wheel p = 18 ÷ 37 and r = 19 ÷ 18 ≈ 1.0556. When r is above 1, rᴺ grows much faster than rᵏ, which is why a distant goal becomes nearly unreachable.

The model assumes independent bets, a constant stake of one unit, even-money payouts and a fixed win probability. Real play with varying stakes, side bets or different payouts needs a different calculation.

Worked Example: Doubling a 100-Unit Bankroll

Suppose a bankroll of k = 100 units, a goal of N = 200 units, one-unit bets on red at an American roulette wheel.

Step 1. Win probability: p = 18 ÷ 38 ≈ 0.473684, so q = 20 ÷ 38 ≈ 0.526316.

Step 2. Ratio: r = q ÷ p = 20 ÷ 18 ≈ 1.111111.

Step 3. Powers: r¹⁰⁰ ≈ 37,648.6 and r²⁰⁰ ≈ 1.4174 × 10⁹.

Step 4. Probability of reaching 200: (1 − 37,648.6) ÷ (1 − 1.4174 × 10⁹) ≈ 37,647.6 ÷ 1,417,400,000 ≈ 0.00002656, or about 0.0027%.

Step 5. Risk of ruin before doubling: 1 − 0.00002656 ≈ 99.9973%.

For comparison, a fair game gives k ÷ N = 100 ÷ 200 = 50%. A smaller goal changes the picture: from 100 units to 110, the American-wheel formula gives about 34.87%, while a fair game gives 100 ÷ 110 ≈ 90.91%. A negative edge hurts more the longer the walk lasts, because the goal requires many more bets than the short target does.

How Bankroll Size and Wheel Type Change the Result

The table shows the probability of doubling the bankroll before ruin with one-unit even-money bets. Fair-game values come from k ÷ N; the others from the edge formula.

Look at the direction of the trend. A larger bankroll measured in bets lowers the chance of doubling in a negative-edge game, because a larger bankroll means more bets must be placed to reach the goal, and every bet carries the same expected loss. In a fair game, the chance stays at 50%.

That does not make larger stakes a better plan. Bigger bets shorten play and raise the chance of a quick double, but they leave the expected loss per dollar wagered unchanged. The Martingale system guide shows the same effect for doubling stakes: it changes the shape of results, not their average.

A positive edge changes the answer. With a hypothetical even-money event that wins 55% of the time and no goal, the ruin probability from k units is (q ÷ p)ᵏ, so from 20 units it is (0.45 ÷ 0.55)²⁰ ≈ 1.8%. Standard casino bets never offer that edge, and in sports markets the true probability is only an estimate, a point developed in the Kelly criterion guide.

Common Mistakes With Risk of Ruin

The first mistake is reading a small ruin number as safety. A 6.28% chance of doubling from 50 units does not mean 93.72% of sessions end in disaster; it means that reaching that specific goal is unlikely. Losing a fraction of the bankroll is far more common than losing all of it.

The second is ignoring the goal. Ruin risk always refers to a stopping rule. Without a goal, a negative-edge game ends in ruin with probability 1 if play continues indefinitely.

The third is using dollars instead of units. A $500 bankroll is 50 units at $10 bets and 10 units at $50 bets, and the two cases have different ruin probabilities even though the dollars are identical.

The fourth is applying a positive-edge formula to a negative-edge game. The infinite-play result (q ÷ p)ᵏ only applies when p is above 0.5; for a casino bet it would give a value above 100%, which is a signal that the wrong formula was used.

Quick Reference and Check Your Answer

Test the formula on k = 1, N = 2. You reach 2 only by winning the first bet, so the probability must equal p. For the American wheel, 1 ÷ (1 + r) = 1 ÷ 2.1111 ≈ 0.4737, which matches 18 ÷ 38. Also check that the result lies between 0 and 1 and equals 50% at p = 0.5 for a goal at double the bankroll.

Related Concepts and Next Steps

Risk of ruin builds on the swings explained in variance in gambling and on the long-run convergence discussed in the law of large numbers guide. The per-bet edge used in the roulette examples is derived in the roulette house edge article.

For stake sizing under a positive estimated edge, see the Kelly criterion guide, which also explains why the correct stake for every standard casino bet is zero. Gambling is best treated as paid entertainment with a known average cost, and ruin probabilities are one way of seeing how that cost accumulates.