What is Monte Carlo simulation?
Monte Carlo simulation is a computational method that uses random sampling to estimate the behavior of a system whose outcomes are uncertain. In the context of betting, it is used to simulate thousands or millions of hypothetical bets to observe how the results distribute, how the bankroll evolves over time, and how the average outcome converges toward the expected value.
The method is named after the Monte Carlo casino in Monaco, reflecting its connection to randomness and probability. It was formalized in the 1940s by Stanislaw Ulam and John von Neumann during their work on nuclear physics at Los Alamos, and has since become a standard tool in statistics, finance, engineering, and any field where probabilistic outcomes must be analyzed.
How it works
A Monte Carlo simulation of a betting scenario operates as follows:
1. Define the parameters of the bet: the decimal odd, the estimated actual probability, and the stake.
2. Generate a random number between 0 and 1.
3. If the random number is less than the actual probability, the bet wins: the profit is stake · (odd − 1).
4. If the random number is greater than or equal to the actual probability, the bet loses: the loss is the stake.
5. Record the result and update the cumulative profit or bankroll.
6. Repeat steps 2 through 5 for the desired number of iterations (for example, 10,000 bets).
After all iterations, the simulation produces a distribution of outcomes that can be analyzed statistically: the average profit per bet, the standard deviation, the maximum drawdown, the probability of bankruptcy, and the growth trajectory of the bankroll.
Demonstrating expected value convergence
One of the most instructive uses of Monte Carlo simulation is to demonstrate the law of large numbers in action. The expected value of a bet is a theoretical quantity — it describes the average outcome over an infinite number of repetitions. In practice, a player places a finite number of bets, and the actual average may deviate substantially from the expected value.
A simulation makes this visible. Consider a bet at a decimal odd of 2.10 with an estimated actual probability of 52%. The expected value is:
e = .52 · (2.10 − 1) − .48 = .52 · 1.10 − .48 = .572 − .48 = .092 = 9.2%
After simulating 100 bets, the average profit per bet might be anywhere between −10% and +30%, because 100 bets is a small sample relative to the variance. After 10,000 bets, the average will almost certainly be very close to 9.2%. The simulation shows this convergence concretely, as a trajectory that fluctuates widely at first and gradually stabilizes around the expected value.
Bankroll trajectory and drawdowns
Monte Carlo simulation can also model the evolution of a bankroll over time, incorporating a specific staking strategy. Rather than a flat stake, the simulation can apply flat percentage staking, Kelly staking, or any other strategy, and track the bankroll after each bet.
This reveals properties that are not apparent from the expected value alone:
— The maximum drawdown: the largest peak-to-trough decline in the bankroll during the simulation. Even with positive expected value, drawdowns of 30%, 50%, or more are possible, depending on the staking strategy and the variance of the bets.
— The probability of bankruptcy: if the bankroll drops to zero or below a minimum viable threshold at any point during the simulation, the player is bankrupt. Monte Carlo simulation can estimate this probability for different staking strategies and bankroll sizes.
— The distribution of final bankrolls: after a fixed number of bets, the simulation produces a range of possible final bankrolls. This distribution shows not just the average outcome but the best case, the worst case, and the spread between them.
Multiple simulation runs
A single simulation run is itself subject to randomness. To obtain reliable statistics, the entire simulation is run many times — for example, 1000 runs of 10,000 bets each. Each run produces a different trajectory due to different random numbers, and the aggregate statistics across all runs provide a robust picture of the expected behavior.
From 1000 runs, a player can determine:
— The average final bankroll across all runs.
— The percentage of runs that ended in bankruptcy.
— The median final bankroll (which may differ substantially from the average if the distribution is skewed).
— The 5th and 95th percentiles of the final bankroll, providing a 90% confidence interval for the bankroll after the simulated number of bets.
Practical applications
Monte Carlo simulation serves several practical purposes for a player:
— Strategy evaluation: Before committing real capital, a player can test different staking strategies under various assumptions about expected value and variance, to determine which strategy produces the best balance of growth and risk.
— Bankroll sizing: The simulation can determine the minimum bankroll required to sustain a given strategy with an acceptable probability of bankruptcy. If the simulation shows a 20% chance of bankruptcy with a ¤500 bankroll but only a 2% chance with a ¤2000 bankroll, the player can make an informed decision about how much capital to allocate.
— Calibrating expectations: By observing the range of possible outcomes, a player can develop realistic expectations about the trajectory of his bankroll and the time required to achieve a target profit. This helps prevent premature abandonment of a sound strategy during a normal variance-induced drawdown.
— Validating probability estimates: If a player tracks his actual results alongside simulated results based on his probability estimates, he can assess whether his estimates are well-calibrated. If actual results consistently fall outside the simulated confidence intervals, it may indicate that the probability estimates are biased.