Gambler’s Ruin Bankroll Survival Probability & Expected Spins to Absorption Calculator

Compute probability of hitting target bankroll N before ruin (0) and expected steps.

Probability of Reaching Goal Before Ruin
25.346 %
Probability of Gambler’s Ruin (0 Units) 74.654 %
Expected Number of Bets Until Absorption 365.25 bets
Fair-Game (p=50%) Baseline Win % 50 %
Inputs
⚡ Scenarios:
units
Starting bankroll in betting units (i).
units
Target bankroll goal in betting units (N > i).
%
Single-trial win probability p % (50% fair coin, 48.65% European roulette, 49.29% craps).
Results & Graph
Probability of Reaching Goal Before Ruin
25.346 %
Probability of Gambler’s Ruin (0 Units)
74.654 %
Expected Number of Bets Until Absorption
365.25 bets
Fair-Game (p=50%) Baseline Win %
50 %

Probability of Reaching Goal Before Ruin

Graph 1 • Gambler’s Ruin Probability • Domain Architecture & Live Telemetry Maps Starting Bankroll Units I, Target Goal Bankroll Units N, Single Bet Win Probability Pct through Gambler’s Ruin Probability Formula.
Probability of Reaching Goal B: 25.346 % Gambler’s Ruin Probability • Invariants & Metrics Probability of Reaching Goal 25.346 % Probability of Gambler’s Rui 74.654 % Expected Number of Bets Unti 365.25 bets Fair-Game (p=50%) Baseline W 50 %
Graph 2 • Probability of Reaching Goal Before Ruin Across Starting Bankroll Units I Sweep (40% → 160%) Evaluates Fair (p=0.5): P_win = i / N; Biased (q/p ≠ 1): P_win = [1 − (q/p)^i] / across 7 operating points (current input highlighted).
0 % 33.19 % 66.38 % 7.04 8units 11.88 12units 17.89 16units 25.35 20units 34.60 24units 46.09 28units 60.34 32units
Graph 3 • Joint Sensitivity: Starting Bankroll Units I × Target Goal Bankroll Units N Compares Probability of Reaching Goal Before Ruin when Target Goal Bankroll Units N is shifted +25% (solid) vs. −25% (dashed).
0 % 41.68 % 83.36 % --- Target Goal Bankroll Units N @ 75% 8.14 -30% 10.83 -15% 14.01 +0% 17.74 +15% 22.12 +30%
Summary: Starting with 20 units toward a 40-unit goal at p = 48.65% per bet gives a 25.35% success probability (74.65% ruin probability) in ~365 expected bets.
Summary: Starting with 20 units toward a 40-unit goal at p = 48.65% per bet gives a 25.35% success probability (74.65% ruin probability) in ~365 expected bets.
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Formula & Mathematical Methodology: Gambler’s Ruin Probability Formula

Fair (p=0.5): Pwin = i / N
Biased (q/p ≠ 1): Pwin = 1 − (q/p)i1 − (q/p)N
Fair (p=0.5): P_win = i / N; Biased (q/p ≠ 1): P_win = [1 − (q/p)^i] / [1 − (q/p)^N]
  • startingBankrollUnitsI: Starting Bankroll Units I (units)
  • targetGoalBankrollUnitsN: Target Goal Bankroll Units N (units)
  • singleBetWinProbabilityPct: Single Bet Win Probability Pct (%)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Gambler’s Ruin Probability

Compute probability of hitting target bankroll N before ruin (0) and expected steps.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: Fair (p=0.5): P_win = i / N; Biased (q/p ≠ 1): P_win = [1 − (q/p)^i] / [1 − (q/p)^N].
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Gambler’s Ruin Probability via Fair (p=0.5): P_win = i / N; Biased (q/p ≠ 1): P_win = [1 − (q/p)^i] / [1 − (q/p)^N].

Frequently Asked Questions

How does the Gambler’s Ruin Bankroll Survival Probability & Expected Spins to Absorption Calculator compute the solution?

It evaluates Fair (p=0.5): P_win = i / N; Biased (q/p ≠ 1): P_win = [1 − (q/p)^i] / [1 − (q/p)^N] directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Game Theory, Ratings & Stochastic Processes

Gambler’s Ruin Bankroll Survival Probability & Expected Spins to Absorption Calculator: inputs, method, and how to read the result

Compute probability of hitting target bankroll N before ruin (0) and expected steps.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Starting Bankroll Units I: Starting bankroll in betting units (i).
  • Target Goal Bankroll Units N: Target bankroll goal in betting units (N > i).
  • Single Bet Win Probability Pct: Single-trial win probability p % (50% fair coin, 48.65% European roulette, 49.29% craps).

What the calculation includes

With the default values shown, Probability of Reaching Goal Before Ruin is 25.346 %. The calculation follows Gambler’s Ruin Probability Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Starting Bankroll Units I from 20 units to 22 units changes Probability of Reaching Goal Before Ruin from 25.346 % to 29.723 %. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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