Draw a circle filling a square sheet of paper. Then close your eyes and poke points randomly, anywhere at all, inside that square. Keep poking, and some points land inside the circle, while others land in the four corners outside it. Remarkably, scattering points purely at random like this alone lets you estimate the value of pi (π) fairly accurately. This way of finding π through probability, instead of calculation, is called the Monte Carlo method.
The principle lies in an area ratio. Draw a circle of radius 1 filling a square with side length 2, and the square's area is 2×2=4, while the circle's area is π×1²=π. So circle's area ÷ square's area = π/4 holds. Now imagine scattering points completely evenly (uniformly) at random inside the square — the probability that a single point lands inside the circle should also get close to this exact same area ratio, π/4.
So flipping the formula around gives π ≈ 4 × (points inside the circle ÷ total points). Scatter just 100 points and the estimate bounces around — 3.0 one time, 3.3 the next — but as you scale up to thousands, tens of thousands of points, the fraction landing inside the circle gets closer and closer to the true probability of π/4, and the estimated π value gets closer and closer to 3.14159.... This is the law of large numbers — "repeat something often enough and it approaches the true probability" — playing out right before your eyes.
That said, the Monte Carlo method has a downside: it converges very slowly. To cut the error down to a tenth, you have to increase the number of points by a hundredfold (this is commonly called the "1/√n rule"). So if you need π accurate to many decimal places, other fast formulas for finding pi (dividing a circle's circumference by its diameter, or methods using infinite series) are far more efficient.
Even so, the Monte Carlo method still matters a great deal, because it truly shines on complex problems where setting up an exact formula is difficult. Questions like how neutrons bounce around inside a nuclear reactor, how a complicated financial product's price will change over time, or what tomorrow's weather will be — these are all nearly impossible to capture with an exact formula. In cases like these, the Monte Carlo method — "randomly simulate a huge number of cases and count the ratio of the results" — becomes, in practice, the only realistic solution available.
On our activity page, you can scatter 100 or 1,000 points inside a square with a single button press, and watch on a graph, with your own eyes, how the estimated π value gets closer and closer to the true π (3.14159...) as the points pile up. Seeing a value that bounced around wildly with just a few points gradually settle down makes it far more tangible why probability and statistics are such powerful tools.