Monte Carlo estimates

Approximate an event by repeated simulated experiments and audit the simulator’s sample space.

13 minutes · advanced · Original Poker4Quant lesson

What you will learn

Explain Monte Carlo error and verify a card simulator against an exact small case.

Prerequisites: Sampling error · Pocket-hand counting

Simulation repeats a model; it does not validate it automatically

A Monte Carlo event estimate is the fraction of simulated trials in which the event occurs. If trials are independent and correctly generated, this is a Bernoulli sample fraction with the same sampling-error mathematics as observed binary data.

The random generator and draw algorithm must implement the stated experiment. For a two-card no-replacement draw, sample a uniform first index and then a uniform remaining index. Sorting a deck by random comparison results is not a justified uniform shuffle algorithm.

Rare events need enough trials and exact checks

An estimate of zero from a finite run does not prove an event is impossible. Rare events can appear irregularly, and reducing ordinary Monte Carlo standard error by half generally requires four times as many independent trials.

Use exact enumeration or a known boundary as a test oracle where feasible. Record the trial count and reproducibility settings. A fixed seed helps debug a pseudo-random implementation, but it is not a mathematical proof that the algorithm samples the right distribution.

Worked example

A simulator estimates the probability of two aces by drawing 10,000 fresh two-card hands. It observes 45 two-ace hands. Compare with the exact model.

The Monte Carlo estimate is 45/10000=0.0045, or 0.45%.
The exact probability is C(4,2)/C(52,2)=6/1326=1/221, about 0.4525%.
An approximate plug-in standard error is sqrt(0.0045×0.9955/10000), about 0.000669.
Agreement within plausible sampling fluctuation is useful, but also inspect the draw algorithm and other event frequencies.

The observed count is compatible with the exact model; it does not prove the simulator correct. Distinguish probability units from percentages when displaying the error.

Hands-on: Repair a sampler, then challenge it with exact events

Construct two conditional card pools, compare replacement with no-replacement simulations and reconcile exact probabilities, observed fractions and plug-in sampling error.

Reading sampler…

Common mistakes

Allowing a repeated card in a no-replacement trial.
Interpreting zero observed events as zero probability.
Treating a reproducible seed as distributional validation.

Keep these ideas

Simulate the defined experiment faithfully.
Report trial count and sampling uncertainty.
Check exact cases rather than trusting plausible-looking output.