Random Sampling

A Monte Carlo simulation estimates something hard to compute exactly by repeating a random experiment many times and averaging the results. The technique needs nothing fancier than random.seed() (for reproducibility — the same seed always produces the same sequence of "random" numbers) and a loop.

import random

random.seed(42)   # same seed -> identical sequence, every run
print(random.random())   # a float in [0, 1)
print(random.uniform(-1, 1))   # a float in [-1, 1)
print(random.gauss(0, 1))       # a draw from a standard normal distribution

Exercises in this course always seed the RNG explicitly — real Monte Carlo code in production wouldn't (you want genuine randomness), but a graded exercise needs a reproducible, checkable answer.

Estimating π by throwing random points at a square is a classic demonstration: a quarter-circle of radius 1 covers π/4 of the unit square's area, so the fraction of random points that land inside it approximates π/4 — multiply by 4 to estimate π itself.

random.seed(0)
count = 0
n = 100_000
for _ in range(n):
    x, y = random.uniform(-1, 1), random.uniform(-1, 1)
    if x*x + y*y <= 1:
        count += 1
print(4 * count / n)   # close to 3.14159..., gets closer as n grows

This is the Law of Large Numbers in action: as n grows, the sample average (fraction of points inside the circle) converges to the true probability. It's also why Monte Carlo error shrinks proportionally to 1/√n — quadrupling your sample count only halves your error.