masrul / MonteCarlo-Pi

Approximating $\pi$ using Monte Carlo sampling

The value of $\pi$ can be obtained using the ratio of area of circle and area of square. Let, a circle with radius $a$ is inscribed inside a square (side = $2a$).

Area of square = $4a^2$
Area of circle = $\pi r^2$

$\pi=4\frac{\mathrm{Area~of~circle}}{\mathrm{Area~of~square}}$

Here, Monte carlo sampling comes handy to approximate the area. We will randomly throw darts aiming a square board which contains a circle. When, we have enough samples, then the area will be proportional to the number of darts on that region. So, $\pi$ can be expressed as following,

$\pi=4\frac{\mathrm{Number~of~darts~inside~circle}}{\mathrm{Total~darts}}$