One of the most famous problems in probability theory is the “Gambler’s Ruin.” It poses a simple question: If a gambler enters a casino with $i$ chips and plays until they either reach $N$ chips (Success) or lose everything (Ruin), what is the probability of each outcome?
Unlike standard “Risk of Ruin” calculators that use standard deviation approximations, our Gambler’s Ruin Calculator uses the exact discrete random walk formulas. It treats every bet as a distinct unit, allowing you to model specific scenarios—including limiting the number of hands played (“Finite Horizon”).
Gambler's Ruin Calculator
Discrete ModelHow to Use the Gambler’s Ruin Calculator
This tool models a “Simple Random Walk with Absorbing Barriers.” Here is how to configure your simulation:
- Starting Units (i): Enter your starting bankroll in units (e.g., 50 chips).
- Target Units (N): Enter your “Walk Away” goal. If you start with 50 and want to double up, enter 100.
- Win Probability (p): The chance of winning a single unit bet.
- 0.500: Fair Coin Flip.
- 0.486: European Roulette (Red/Black).
- 0.493: Blackjack (Basic Strategy).
- Enable Finite Horizon (Optional): Check this box if you want to know the probability of ruin within a specific number of bets. If uncheck (Infinite Horizon), the calculator assumes you play forever until you hit 0 or N.
Real-World Examples: The Power of the House Edge
The math behind Gambler’s Ruin reveals why casinos always win in the long run, even with a tiny edge.
Example 1: The Fair Game (50/50)
You start with $50 and want to reach $100. The game is a fair coin flip ($p = 0.5$).
- Probability of Ruin: 50%.
- Probability of Success: 50%.
- Expected Duration: 2,500 bets.
- Analysis: In a fair game, your chance of doubling your money is exactly proportional to your starting stack.
Example 2: The Roulette Reality
You start with $50 and want to reach $100 playing Red/Black on European Roulette ($p = 0.486$). The difference seems small, right?
- Probability of Ruin: ~74%.
- Probability of Success: ~26%.
- Analysis: A small 2.7% house edge drastically shifts the odds. You are almost 3 times more likely to go bust than to double up.
Example 3: Finite Horizon (Time Limits)
You have an advantage ($p = 0.51$, card counting), but you only have time for 100 hands.
- Infinite Horizon Success: High (eventually, the edge pays off).
- Finite Horizon (100 Steps): Low.
- Analysis: Even with an advantage, 100 steps is often not enough time to reach a high target. The calculator will show a large “Ongoing Chance,” meaning the session likely ends somewhere in the middle.
Frequently Asked Questions (FAQ)
What is an “Absorbing Barrier”?
In a random walk, an absorbing barrier is a state that, once reached, cannot be left. In gambling, 0 (Bankruptcy) and N (Target Goal) are absorbing barriers. Once you hit either, the game stops.
What is the difference between Infinite and Finite Horizon?
Infinite Horizon assumes you have unlimited time and will play until you win or bust. Finite Horizon calculates the probabilities for a specific number of steps (e.g., “What are my chances of ruin in the next 500 hands?”).
Does bet sizing affect Gambler’s Ruin?
Yes. This calculator uses “Units.” If you have $1,000 and bet $10, you have 100 Units. If you bet $100, you have 10 Units. Generally, betting smaller units decreases your risk of ruin but drastically increases the time required to reach your goal.
Why is “Expected Duration” important?
It tells you the average number of bets the session will last. For professional gamblers, this is crucial for calculating hourly earnings. If a game takes 10,000 steps to reach the target, but you can only play 50 hands an hour, the “Success” might not be worth the time investment.