Field · 6 models
Probability Simulations
Probability questions often have exact answers that feel wrong. These simulators run the experiment thousands of times and put the observed frequency next to the calculated probability, so intuition and arithmetic can be compared directly.
Models in this field
- OS-006Coin flipFlip a fair or biased coin: the share of heads settles down, and long streaks keep happening.
- OS-007Dice rollRoll k dice with s sides and check the histogram of sums against the exact distribution.
- OS-008Birthday paradoxWhy 23 people give a better-than-even chance that two of them share a birthday.
- OS-009Monty Hall problemStay or switch? Thousands of games show when switching wins — and why the host’s knowledge matters.
- OS-010Galton boardBalls bouncing left or right through rows of pegs pile up into a binomial bell curve.
- OS-011Central limit theoremAverage n draws from almost any distribution and the means turn normal — at a speed you can measure.
Simulation as a check, not a replacement
Every model in this field has a closed-form answer: the binomial distribution for coins and the Galton board, a convolution for dice sums, a product formula for shared birthdays, simple fractions for Monty Hall. The simulations show the answer being approached, how noisy small samples are, and where the assumptions behind the formula live — independent flips, uniform birthdays, a host who knows where the car is.
Suggested order
- Coin flip: the law of large numbers, and why streaks are longer than people expect.
- Dice roll: from a flat distribution for one die to a bell shape for the sum of several.
- Galton board: a physical picture of the binomial distribution.
- Central limit theorem: why sums and averages of almost anything look normal, and how fast.
- Birthday paradox and Monty Hall: two results that surprise almost everyone the first time.
Frequency and probability
A simulated frequency is an estimate of a probability, not the probability itself. After 100 fair coin flips, a heads share of 0.45 is unremarkable; after 100,000 it would be strong evidence of a biased coin or a broken generator. Each page shows the standard error so you can tell which is which. When the model is right, the gap between frequency and probability shrinks like 1/√n — the same rate that governs every Monte Carlo estimate.