A probability is a long run, not a promise

The probability of an outcome is the proportion of times it would occur in an unending sequence of repetitions of a random process, carried out under the same conditions. In symbols, if the outcome occurs $k$ times in $n$ trials, then

$$\hat{p} = \frac{k}{n} \longrightarrow P(\text{outcome}) \text{ as } n \text{ grows}.$$

Two consequences do most of the work in this unit. Every probability lies between 0 and 1, because a count of successes cannot be negative or exceed the number of trials. And a probability describes the long run only: it makes no claim about the next trial, or the next ten. A fair coin landing heads six times running is unusual, not impossible, and the coin carries no memory of it.

Designing a simulation is four decisions, and the first is the one that goes wrong:

1. Model the chances. Choose a device (random digits, a spinner, cards, a random number generator) and assign outcomes so that each has exactly the intended probability. For a 30% event using digits 0 through 9, three of the ten digits mean success: 0, 1, 2. Letting 0 through 3 count as success models 40%, not 30%.
2. Define one trial. State precisely what one repetition consists of and when it ends: "draw five digits, one per free throw", or "draw digits until a success appears, then stop".
3. Define the outcome of interest. Say what counts as a success for the trial as a whole, matching the question's wording exactly, including "at least" and "at most".
4. Repeat many times and count. Run a large number of trials and report the proportion of trials that produced the outcome.

Two design details separate a correct simulation from a plausible one. Ignoring versus reusing: to model a 6-outcome situation with digits, digits 7, 8, 9 and 0 are discarded, not folded onto other outcomes, because folding them changes the probabilities. And sampling without replacement means a value already used cannot appear again in the same trial, which the design must enforce.

The law of large numbers states that as the number of trials grows, the relative frequency of an outcome approaches its probability. It is a statement about a limit, and it is what makes simulation work at all.

What it does not say is that results compensate. After six heads in a row, the probability of heads on the seventh toss is still 0.5: the coin has no record of the first six. The long-run proportion drifts back toward 0.5 not because tails are owed, but because the six get diluted by thousands of later tosses. Expecting a correction is the gambler's fallacy, and it appears in every disguise the topic offers: a slot machine "due", a basketball shooter who "has to miss soon", a simulation stopped early because the results "have not evened out yet".

The same reasoning is what makes a short run uninformative. Seven heads in eight tosses is entirely ordinary for a fair coin, so it is no evidence of unfairness. Judging a claim by a handful of trials is the error; the whole point of running many is that only many can separate a real difference from ordinary variability.

Because a simulation is itself a random process, its result is an estimate that would come out differently on the next run. Two simulations of the same situation, each with 100 trials, might report 0.31 and 0.37, and neither is the true probability.

More trials means less variability in the estimate. The typical distance between $\hat{p}$ and the truth shrinks roughly in proportion to $\frac{1}{\sqrt{n}}$, so getting an estimate that is twice as precise takes about four times as many trials. Reporting more decimal places than the run supports is a claim the simulation cannot back.

The vocabulary belongs in every write-up: "in 1,000 simulated seasons, 214 produced at least three injuries, so the estimated probability is 0.214". The word estimated is doing real work. When an exact computation is available, the simulation is a check on it; when it is not, the simulation is the answer, with its uncertainty attached.