Darts in the Unit Square

When a number is hard to calculate exactly, Monte Carlo simulation generates lots of random examples and lets the pattern in them reveal it. Here the number is $\pi$.

Throw darts uniformly at random into the unit square, $0 \le x < 1$ and $0 \le y < 1$. A dart lands inside the quarter circle of radius $1$ when $x^2 + y^2 \le 1$. The fraction of darts that land inside settles toward the quarter circle's share of the square's area — and from that fraction you can estimate $\pi$.

Task: write estimate_pi(seed, checkpoints), which throws darts one at a time and returns the running estimate of $\pi$ at each checkpoint.

• Make the random numbers with rng = random.Random(seed). For each dart, call rng.random() twice: the first value is $x$, the second is $y$.
• checkpoints is a list of dart counts in increasing order, e.g. [10, 100, 1000]. At each checkpoint, report the estimate based on every dart thrown so far.
• Throw exactly as many darts as the last checkpoint — the judge replays the same darts from the seed.
• Return one estimate per checkpoint, each rounded to 4 decimal places.

Watch the estimates as the checkpoints grow. Their drift toward $3.1416$ is the Law of Large Numbers at work, and how slowly they get there — the error shrinks only like $1/\sqrt{N}$ — is why Monte Carlo needs so many samples.