Gambler's Ruin Probability Calculator

Formulas

The standard difference-equation solution yields a closed form. There are two cases: a fair game (p = 1/2) and a biased game (p ≠ 1/2).

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

When wins and losses are equally likely, the probability of reaching N before 0 is linear in the starting capital:

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).

Probability of ruin

Once you have P(success): P(ruin) = 1 − P(success).

How to interpret the result

• Higher i (initial capital) increases your success probability, because you start farther from ruin.
• Higher N (target) decreases your success probability, because you must accumulate more net wins before a net run of losses takes you to 0.
• p above 0.5 can improve your odds substantially, but it still does not guarantee success unless p = 1. With finite capital and a finite target, there is still meaningful ruin risk.
• p below 0.5 makes success increasingly unlikely as N grows.

A useful intuition: for p ≠ 1/2, the expression depends on powers of q/p. Exponentials change quickly, so small differences between p and 0.5 can lead to large differences in the probability when N is large.

Worked example

Suppose you start with i = 10 units and your goal is N = 50 units.

Example A: Slightly unfavorable game (p = 0.49)

Here q = 0.51 and r = q/p = 0.51/0.49 ≈ 1.040816. Because r > 1, the powers rⁱ and r^N grow, and the success probability becomes small. Using the biased formula:

P(success) = (1 − r¹⁰) / (1 − r⁵⁰).

Numerically this is only a modest chance of hitting 50 before ruin, despite starting at 10.

Example B: Slightly favorable game (p = 0.51)

Now q = 0.49 and r = 0.49/0.51 ≈ 0.960784. Because r < 1, the powers decay with larger exponents, boosting success probability compared with the unfavorable case.

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

The key takeaway is not the exact decimals in this write-up, but the sensitivity: moving p from 0.49 to 0.51 can dramatically change the outcome when the target N is far away.

Assumptions & limitations

• Unit step size: each round changes wealth by exactly +1 or −1. If you vary bet sizes, the model changes.
• Fixed win probability: p is constant and does not depend on time, your fortune, or strategy.
• Independent rounds: outcomes are independent from round to round (no streak dependence).
• Absorbing barriers at 0 and N: play stops immediately upon reaching 0 (ruin) or N (target).
• Integer framing: the classic derivation assumes integer states 0,1,2,…,N. Non-integer “units” require a re-interpretation or rescaling.
• Target vs. starting capital: typically you assume 0 < i < N. If i ≥ N, success is already achieved (probability 1). If i ≤ 0, ruin is already realized (probability 0).
• Edge probabilities: if p = 0 then success probability is 0 (unless i ≥ N); if p = 1 then success probability is 1 (unless i ≤ 0).
• Not a betting system validator: this model does not incorporate stop-loss rules, table limits, changing odds, or bankroll management schemes.

Quick comparison table

Model setup

Let X_t be your fortune after t rounds. You start at X₀ = i with two absorbing barriers: 0 (ruin) and N (target). On each round:

• 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).