Estimating Probabilities Using Simulation

Probability is the long-run relative frequency of an outcome, $\hat{p} = \frac{k}{n}$ approaching $P$ as $n$ grows, which is exactly the fact simulation exploits. A correct simulation assigns outcomes to a random device so the modeled chances match the real ones (three digits out of ten for a 30% event, with unused digits discarded rather than folded in), defines one trial and its stopping rule, states what counts as a success in the wording of the question, then repeats many times and reports the proportion. The law of large numbers guarantees that this proportion approaches the probability, and the estimate's typical error shrinks like $\frac{1}{\sqrt{n}}$.

The failures cluster on the short run. Streaks are read as debts: six heads and tails is "due", though the seventh toss is still 0.5, since the long-run proportion recovers by dilution and not by compensation. Small runs are read as verdicts: seven heads in eight tosses declared proof of an unfair coin, or a 20-trial simulation reported as the probability rather than as a noisy estimate. And the design itself quietly changes the question, most often by assigning four digits of ten to a 30% event or by reusing digits that should have been discarded, so that thousands of correctly executed trials answer a situation nobody asked about.