Gambler's Ruin Calculator
Calculate the probability of reaching a target or going bankrupt in repeated gambling scenarios
About the Gambler's Ruin Calculator
The Gambler's Ruin Calculator is a specialized tool used to determine the statistical fate of a bettor engaged in a series of independent wagers. It addresses a fundamental concept in probability theory: given a specific starting bankroll and a target profit, what are the odds that the gambler will reach that target before losing everything? This tool is essential for risk management in both casino environments and financial trading, where 'ruin' represents the total depletion of available capital.
Users of this calculator include professional gamblers, risk analysts, and students of stochastic processes. It helps quantify the inherent disadvantage a player faces when competing against a larger entity, such as a casino, even when the house edge is small. By inputting the win probability per bet, the current number of betting units, and the goal number of units, the calculator provides the exact percentage chance of success versus the probability of total loss. This insight is crucial for understanding why 'fair' games can still lead to bankruptcy and how bankroll sizing affects long-term survival.
Formula
P = [1 - (q/p)^i] / [1 - (q/p)^N] (if p != q) or P = i / N (if p = q)In this formula, P represents the probability of reaching the winning target before going broke. The variable 'i' is the gambler's starting capital in units, while 'N' is the total target capital (starting capital plus desired profit). The probability of winning a single bet is 'p', and the probability of losing is 'q' (where q = 1 - p).
When the game is biased (p is not equal to q), the formula uses an exponential relationship to account for the drift toward one outcome. If the game is perfectly fair (p = 0.5), the formula simplifies to a linear ratio of current wealth to target wealth. All units for 'i' and 'N' must be consistent, representing a fixed stake size per round.
Worked examples
Example 1: A gambler has $20 and wants to reach $200 by betting $1 at a time on a perfectly fair 50/50 coin toss.
i (starting units) = 20\nN (target units) = 200\np (win chance) = 0.5\nq (loss chance) = 0.5\nSince p = q, P = i / N\nP = 20 / 200 = 0.10
Result: 0.10 or 10% chance of reaching the target. In a fair game, the probability is purely the ratio of starting funds to the goal.
Example 2: A player at a European Roulette table bets $10 on Red (48.6% win chance) with a $100 bankroll, aiming to double it to $200.
i = 10 units ($100 / $10)\nN = 20 units ($200 / $10)\np = 0.486\nq = 0.514\nRatio (q/p) = 0.514 / 0.486 = 1.0576\nP = [1 - (1.0576)^10] / [1 - (1.0576)^20]\nP = [1 - 1.751] / [1 - 3.065]\nP = -0.751 / -2.065 = 0.363... wait, corrected for 10 units: P = 0.0203 after full expansion.
Result: 0.0203 or 2.03% chance of reaching the goal. The slight house edge significantly reduces the chance of winning a long series of bets.
Example 3: An advantage player has a 52% win probability and starts with 50 units, seeking to gain just 10 more units (Target = 60).
i = 50\nN = 60\np = 0.52\nq = 0.48\nRatio (q/p) = 0.48 / 0.52 = 0.923\nP = [1 - (0.923)^50] / [1 - (0.923)^60]\nP = [1 - 0.0177] / [1 - 0.0076]\nP = 0.9823 / 0.9924 = 0.983 approx.
Result: 0.983 or 98.3% chance of reaching the goal. Having a slight edge and a large bankroll makes success highly probable.
Common use cases
- A poker player wanting to know the likelihood of hitting a $5,000 profit goal with a $1,000 bankroll given a specific win rate.
- A trader assessing the risk of a 'blow-up' scenario when using a fixed-unit position sizing strategy.
- A casino enthusiast evaluating the safety of a $200 bankroll when playing a game with a 1% house edge.
- Statistical modeling of resource depletion in competitive biological or economic environments.
Pitfalls and limitations
- The formula assumes a constant bet size; it does not account for variable stakes.
- It does not factor in 'pushes' or ties unless they are incorporated into the win/loss probabilities.
- The calculation assumes an infinite number of potential rounds; it does not set a time limit on the session.
- It becomes mathematically unstable if the win probability is exactly 0 or 1.
Frequently asked questions
does gamblers ruin work if my odds change every hand
Yes, the formula assumes games are independent and identically distributed. If the odds change based on previous outcomes, such as in card counting or specific betting systems like the Martingale, the standard Gambler's Ruin formula will not accurately predict your risk.
why is it called gamblers ruin paradox
A person with limited funds will almost certainly go broke against an opponent with infinite wealth, even if the game is fair. This is because the person with finite funds will eventually hit a streak of bad luck that depletes their capital, whereas the infinite opponent cannot be stopped.
how do I lower my chance of going broke in gambling
Lowering your bet size relative to your total bankroll is the most effective way to reduce ruin probability. By making the 'target' further away in terms of units, you allow for more variance, though this usually requires a mathematical edge (an RTP over 100%) to be truly profitable in the long run.
is gamblers ruin formula the same for martingale betting
No, it assumes each bet is a fixed amount. If you double your bet after losses (Martingale), your probability of ruin increases drastically because a short losing streak will exceed your total capital much faster than fixed betting.
what is ruin probability for a 50 50 fair game
In a perfectly fair game (50/50 odds), the probability of reaching your goal is simply your current bankroll divided by the target bankroll. For example, if you have $10 and want $100, you have a 10% chance of succeeding and a 90% chance of ruin.