Estimating π with Random Darts

Throw darts at random into a square with a quarter circle inside. The chance a dart lands inside is the area ratio, π/4, so four times the fraction that do estimates π, and the estimate tightens as the darts pile up.

How it works

Probability as long-run relative frequency

In one sentence

The probability of an event is the long-run relative frequency of its occurrence. For a point chosen uniformly in the unit square, $P(x^2 + y^2 \le 1) = \pi/4$ , and by the law of large numbers the proportion of hits $h/n$ converges to it as $n \to \infty$ .

General formula

{\htmlData{sym=0}{P(\text{inside})}} = \frac{\text{area of quarter circle}}{\text{area of square}} = \frac{\pi}{4} \approx \frac{\htmlData{sym=1}{{h}}}{\htmlData{sym=2}{{n}}}

Random points sprinkle the square evenly, so counting how many land in a region measures its area.

What the symbols mean

$P(\text{inside})$
Probability of a hit
The chance one random dart lands inside the quarter circle.
$h$
Hits
How many darts landed inside the quarter circle.
$n$
Darts thrown
The total number of random points.

Each dart lands at a random point, equally likely anywhere in the square. That makes the probability of an event equal to its share of the area.
The quarter circle has area $\pi/4$ and the square area 1, so $P(\text{inside}) = \pi/4$ .
By the law of large numbers the fraction of hits $h/n$ approaches that probability as $n$ grows, so $4h/n$ approaches $\pi$ .
The error shrinks only like $1/\sqrt{n}$ : a hundred times more darts buys about one more correct digit.

Learn the math

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