Gambler's ruin · the math of going broke

Even a fair game takes everything, eventually.

A finite bankroll against an endless house. Run the random walks — watch every line drift down to zero, and watch the closed-form ruin probability call it before the dice are thrown.

Analytic ruin probability

100.0%

P(hit 0 before goal), Feller Ch. XIV

1 − N/M

Empirical bust rate

—

0 busted · 0 reached goal · 0 runs

Bankroll over time · each line is one bettor —

busted reached goal still alive at cap goal · zero lines

Run simulations

Hit zero

Hit goal

Setup

Starting bankroll₱10,000

Bet size₱500

Win probability50%

Fair coin — zero edge. Ruin is still certain.

Goal (target bankroll)₱20,000

Max flips per trajectory2,000

How to use

Set starting bankroll and bet size, then hit Run. Each trajectory plays a coin-flip random walk until it hits zero, reaches the goal, or runs out of flips. The card up top compares the analytic ruin probability against the measured bust rate — they converge as you run more.

What the theorem says

Gambler's ruin: a player with a finite bankroll betting against an infinite-bankrolled opponent (the house) will go to zero with probability 1, given enough time, even at fair (50/50) odds.

With unfair odds (house edge), ruin probability is even higher and faster.

With win probability p, lose probability q = 1−p, starting bankroll N, target M (in bet-units), the probability of reaching M before ruin is:

P(reach goal) = (1 − (q/p)ᴺ) / (1 − (q/p)ᴹ) when p ≠ q
P(reach goal) = N / M when p = q = ½

Ruin probability is just 1 minus that. Here N and M are the bankroll and goal measured in bet-units (bankroll ÷ bet).

Why this matters

Real casinos don't offer fair coins. Roulette, slots, sports books — all carry 2–15% house edges. With a small edge against you and many bets, the math is decisive: you bust before you reach any meaningful target.

Try setting win probability to 47% (roughly a single-zero roulette red bet). Almost every trajectory busts, and quickly.

The only winning strategy in a negative-EV game is not playing. Or playing very rarely with small fixed amounts as entertainment.

Variance is the trap

Trajectories that hit zero quickly are boring. The dangerous ones are those that climb high before crashing — they confirm your "system" is working, increase your bet size, then ruin you.

Watch the chart: even at fair odds, some lines double or triple before coming back. Each one feels like winning. Each one ends at zero.

Citations

Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1 (3rd ed.). Wiley. Ch. XIV — the canonical treatment of gambler's ruin and absorbing-state random walks.
Huygens, C. (1657). De Ratiociniis in Ludo Aleae. The earliest known formal statement of the gambler's-ruin problem.
Karlin, S. & Taylor, H.M. (1975). A First Course in Stochastic Processes (2nd ed.). Academic Press. Ch. 3 on Markov chains and absorption probabilities.
Doob, J.L. (1953). Stochastic Processes. Wiley. Martingale framing of the ruin problem.

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