Monte Carlo accuracy and sampling error

Monte Carlo simulation estimates a probability by repeating randomly sampled legal deals. If a hand wins 6,240 of 10,000 trials, the observed win rate is 62.4%. A second run will usually differ slightly because it samples a different set of legal completions. That variation is expected measurement noise, not automatically a defect.

Standard error for a simulated proportion

Here p is the underlying event probability and n is the number of independent trials. The largest standard error occurs near p = 0.5, so 50% provides a conservative reference. A rough 95% sampling margin is 1.96 times the standard error. This interval describes random trial variation under the model; it does not measure whether the model itself represents a real opponent.

Why ten times more trials is not ten times tighter

Error shrinks with the square root of n. Reducing random error by half requires roughly four times as many trials; reducing it to one tenth requires about one hundred times as many. This creates a practical tradeoff: extra precision costs computation, while the displayed decimal places may imply more certainty than the sample supports.

Wins, ties, and losses share the same trials

The three displayed outcomes are not separate experiments. Every trial contributes to exactly one win, tie, or loss category, so their estimates are linked and should total 100% before display rounding. A rare tie rate can jump proportionally between small runs even when the win rate looks stable. Save raw trial counts when testing the implementation; rounded percentages hide useful detail.

Convergence is a diagnostic, not proof

Run the same fixed scenario at increasing trial counts and record each result. Large swings that persist at high counts can reveal a dealing or evaluation bug. Smooth convergence only shows that the program repeatedly samples its own model. A perfectly converged result can still be irrelevant if it assumes uniformly random opposing cards when the question requires a specific weighted range.

Closed-form checks are stronger when available

Some events have exact combinatorial answers. A nine-out flush draw after the flop completes by the river with probability 1 − (38/47 × 37/46), approximately 34.97%. A simulator using the same card-removal conditions should approach that value as trials increase. Comparing sampled output with an exact result tests more than comparing two random runs with each other.

A practical validation protocol

1. Freeze every known card, opponent count, and configuration value.
2. Run several independent samples at each available precision level.
3. Record raw counts, percentages, runtime, and the random seed if one is available.
4. Compare the spread with the expected sampling error.
5. Use an exact probability check for at least one simplified scenario.