Gambler's Ruin Probability Calculator

Gambler's ruin formulas

The gambler's ruin model has a direct solution, so this calculator can report the success probability without running a random trial. The expression depends on whether the round-to-round odds are fair (p = 1/2) or biased (p ≠ 1/2).

Case 1: Fair game (p = 1/2)

When wins and losses are equally likely, the probability of reaching N before 0 grows linearly with the starting bankroll:

Case 2: Biased game (p ≠ 1/2)

Let q = 1 − p. Define the ratio r = q/p. Then:

P(success) = (1 − rⁱ) / (1 − r^N).

Written directly in terms of p and q: P(success) = (1 − (q/p)ⁱ) / (1 − (q/p)^N).

In a favorable game, q/p falls below 1 and the success probability bends upward as the bankroll gets closer to the target. In an unfavorable game, q/p rises above 1 and the powers work against the player as the goal gets farther away.

Probability of ruin

Once you have P(success), the ruin probability is its complement: P(ruin) = 1 − P(success).

That relationship is especially useful when you want to talk about downside risk instead of upside chance. A result near 1 for P(ruin) means the bankroll is much more likely to disappear before the target is reached, while a result near 0 means the target is relatively close in model terms.

How to interpret gambler's ruin probabilities

• Higher i (initial capital) gives the bankroll a larger cushion, so the success probability increases.
• Higher N (target) makes the finish line farther away, so the success probability decreases.
• p above 0.5 gives the walk a favorable edge, but it still does not guarantee success unless p = 1. Even with an advantage, a bankroll can still hit 0 before it reaches the target.
• p below 0.5 makes the upward path harder to sustain, so success becomes less likely as N grows.

In the biased formula, the ratio q/p is the main driver of the result. When that ratio is less than 1, larger exponents help the player; when it is greater than 1, larger exponents work in the opposite direction. That is why a small change in p can matter a great deal when the starting bankroll is modest or the target is distant.

Another way to think about the output is to treat it as a model-based survival chance for the bankroll. The calculator does not predict the next hand or the next spin; it summarizes the long-run structure of the walk from i toward N with a hard stop at 0.

Worked gambler's ruin example

Suppose the bankroll starts at i = 10 units and the goal is N = 50 units.

Example A: Slightly unfavorable gambler's ruin setup (p = 0.49)

Here q = 0.51 and r = q/p = 0.51/0.49 ≈ 1.040816. Because r > 1, each extra step toward the target pushes the powers in the wrong direction for the player, and the success probability is pulled downward. Using the biased formula:

P(success) = (1 − r¹⁰) / (1 − r⁵⁰).

In practical terms, the slight house edge compounds over time. Even though the starting bankroll is not tiny, the model still says the target is separated from the starting point by enough distance for losses to matter quickly.

Example B: Slightly favorable gambler's ruin setup (p = 0.51)

Now q = 0.49 and r = 0.49/0.51 ≈ 0.960784. Because r < 1, the powers decay as the exponents grow, so the success probability is stronger than in the unfavorable case.

P(success) = (1 − r¹⁰) / (1 − r⁵⁰), with a meaningfully higher result than in Example A.

The point of the example is the sensitivity of the gambler's ruin model: moving p from 0.49 to 0.51 flips the edge from slightly negative to slightly positive, and that shift can have a large effect once the target is far enough away.

If you want to test that sensitivity yourself, keep the starting capital fixed and compare nearby probabilities. In ruin problems, a small change in the round-level edge often matters more than a small change in the target when the bankroll is already close to either boundary.

Gambler's ruin assumptions & limitations

• Unit step size: each round changes wealth by exactly +1 or −1. If you vary bet sizes, the classic formula no longer applies as written.
• Fixed win probability: p stays constant and does not depend on time, the current bankroll, or any betting system.
• Independent rounds: one round does not alter the next, so streaks have no built-in memory.
• Absorbing barriers at 0 and N: the walk stops immediately upon reaching 0 (ruin) or N (target).
• Integer framing: the textbook derivation uses whole-number states 0, 1, 2, …, N, so non-integer stakes need a rescaling or reinterpretation.
• Target vs. starting capital: this setup normally assumes 0 < i < N. If i ≥ N, the target has already been reached; if i ≤ 0, ruin has already occurred.
• Edge probabilities: if p = 0, success is impossible unless i ≥ N; if p = 1, success is certain unless i ≤ 0.
• Not a betting system validator: the model does not include stop-loss rules, table limits, changing odds, or bankroll-management overlays.

Gambler's ruin probability comparison table

This comparison table summarizes the three most common gambler's ruin cases so you can see how the formula changes when the game is fair, favorable, or unfavorable.

Gambler's ruin model setup

The model behind this calculator treats the bankroll as a one-dimensional walk between 0 and N, with the current balance at round t written as X_t. You start at X₀ = i, and the two absorbing barriers are the ruin point at 0 and the target point at N.

• With probability p, X increases by 1.
• With probability q = 1 − p, X decreases by 1.

The key quantity is the success probability: P(success) = P(reach N before 0 | start at i). The ruin probability is simply P(ruin) = 1 − P(success). That is the value the calculator reports, along with the complement, so you can judge both the upside and the downside in one glance.

In gambling terms, the state space is the bankroll ladder from 0 to N. In probability terms, the calculator is answering a first-passage question: which absorbing boundary gets hit first? That framing is what makes the gambler's ruin model so useful for bankroll planning, even when the real-world situation is more complicated than a one-unit walk.