Estimation of pi by Monte Carlo method

How does it work?

Points are generated randomly and follow a uniform distribution in the square.

Hence, the probability that a point is generated in the circle is

$\mathcal{P} = \frac{\mathcal{A}_{circle}}{\mathcal{A}_{square}} = \frac{\pi r^2}{(2r)^2}=\frac{\pi}{4}$

If we call $p_n$ the proportion of points inside the circle when we have generated $n$ points, according to the law of large numbers the limit of $4 p_n$ is

$\lim\limits_{n \to +\infty} 4 p_n = 4 \mathcal{P}=\pi$

So we use $4 p_n$ as an estimation of $\pi$, and it becomes a good estimation when there are enough points.