Monte Carlo Simulation: Learn Probability by Repeating the Model

A Monte Carlo simulation estimates properties of a model by repeatedly sampling from it. To estimate a probability, define an event, simulate many trials and divide the number of successes by the number of trials. The procedure can make abstract probability visible, but it only answers the question encoded in the model.

Simulating a fair lottery is not the same thing as predicting an actual draw.

Define one trial before writing a loop A trial might be one complete 6/49 draw, a pair of independent draws, or a 100-draw training block followed by a 20-draw test block. These units have different dependence structures. Counting every ball or every overlapping window as a separate independent trial can understate uncertainty.

For a consecutive-pair experiment, one trial is a complete uniformly sampled six-label set. Success means at least one adjacent difference equals one after sorting. That definition is precise enough for another reader to implement independently.

Estimate and quantify uncertainty If N independent trials produce S successes, the estimate is p-hat = S/N. Away from extreme probabilities and with adequate counts, an approximate standard error is √[p-hat(1−p-hat)/N]. Confidence procedures need extra care near zero or one and when trials are dependent.

A simulation output with six decimal places does not necessarily have six reliable decimal places. The number of trials and event rarity determine the Monte Carlo uncertainty.

Check against something exact For at least one consecutive pair in a k-of-n draw, the exact probability is 1−C(n−k+1,k)/C(n,k). That gives a strong benchmark for the simulation. A result that differs slightly can be ordinary sampling variation; a systematic discrepancy across large independent runs may reveal a coding error.

Also check invariants: each line has k distinct labels in range, category counts sum to N, and observed totals respect the fixed number of selections. These checks target meaningful failure modes rather than merely repeating the implementation.

Record the full method Publish the seed, generator, environment, parameters and code. State which data are raw and which are aggregates. A compact CSV of category counts can be sufficient to inspect a published figure, while a seed and program allow the original trials to be recreated.

The Lab uses a named seeded PRNG for reproducibility. It does not call those streams “truly random” or suggest that the seed has predictive value.

Know when simulation is the wrong tool An exact combination count is usually better for a jackpot probability. A million trials gives very little information about a one-in-hundreds-of-millions event. A smaller model can illustrate the logic more effectively, provided it is clearly labelled as a smaller model.

The browser simulator focuses on common measurable properties such as adjacency and mean sums. The published experiments handle larger runs and retain their results. Both help explore consequences of assumptions; neither turns those assumptions into evidence that a future lottery result can be forecast.