Simulation vs Probability

A simulation answers a probability question by generating outcomes from a model you have assumed and then counting. Fix the model, define exactly what one trial is, run many trials, and report $\hat{p}$ (p-hat) as successes divided by trials. What comes back is an estimate of the probability under that model. It is not the probability itself, and it is not evidence the model is true.

Take a player who makes 70 percent of her free throws with shots independent, and ask for the chance she makes all four of her next attempts. The exact answer is $0.70^4 = 0.2401$. One run of 20 simulated trials gave 3 successes, an estimate of 0.15. A separate run of 10,000 trials from the same model gave 2,335 successes, or 0.2335. Same model both times, and the longer run lands closer.

The misreading: my simulation gave 0.15, so the probability is 0.15. An estimate from $n$ trials has standard deviation $\sqrt{p(1-p)/n}$, which at $p = 0.24$ and $n = 20$ is 0.096. Two standard deviations either side of 0.24 spans roughly 0.05 to 0.43, so a 20-trial run landing on 0.15 is unremarkable. Reporting it as the answer treats a number with that much slop as exact.

Precision improves slowly, and the square root is why. The standard deviation falls with $\sqrt{n}$, so halving the error takes four times the trials and cutting it to a tenth takes a hundred times: 0.043 at 100 trials, 0.0043 at 10,000.

The harder limit is the model. That simulation tells you what a 70 percent shooter would do. It cannot tell you whether this player is a 70 percent shooter, and if she is really at 0.60 then every figure it produced answers the wrong question. Topic 2.3 in the Fall 2026 course is Estimating Probabilities Using Simulation.

Every probability is a number in the interval from 0 to 1, and the probabilities of all the outcomes in the sample space add to exactly 1. $P(A)$, read "the probability of A", is the long-run relative frequency of the event $A$: the share of trials on which $A$ happens, once the number of trials is large. When every outcome is equally likely you can reach that number by counting, dividing the outcomes in $A$ by the total, but counting is a special case rather than the definition.

Draw a single digit at random from 0 through 9, each with probability 0.1. Let $A$ be "the digit is 0, 1, or 2". Then $P(A) = 3/10 = 0.30$.

"A probability of 0.30 means three out of every ten." Read ten digits from a random digit table and the chance that exactly three of them fall in $A$ is 0.2668, so that reading is wrong about 73 percent of the time, and roughly 2.8 percent of ten-digit runs contain none from $A$ at all. A probability is a rate the results approach over a long run, not a quota each block of ten has to fill. What the long run does and does not promise is the content of the law of large numbers.

Two boundaries are worth naming. Counting outcomes gives the right answer only when the outcomes really are equally likely, which rules out a loaded die or a thumbtack. And probability 0 does not always mean impossible: for a continuous random variable every single exact value has probability 0, yet the variable still lands somewhere.

Probability is introduced in topic 2.4, Introduction to Probability, after simulation has already given you an empirical version of the same quantity.