Simulation and the Law of Large Numbers

Intuition

The expected sum of two fair dice is 7. Roll them 10 times and the average may be 6.2 or 8.1; roll them 100 000 times and it will be very close to 7. The law of large numbers says sample averages settle down to the expected value as the sample grows.

Precise explanation

For i.i.d. $X_i$ with mean $\mu$ , the running mean $\bar X_n = \frac1n\sum_{i\le n} X_i$ converges to $\mu$ . Its typical error shrinks like $\sigma/\sqrt n$ : 100× more rolls ≈ 10× less error.

Reproducible randomness. random.Random(seed) creates an independent generator. Python only guarantees the sequence of rng.random() (floats in $[0, 1)$ ) across versions, so we derive die faces from it:

\text{face} = \lfloor 6u \rfloor + 1,\quad u = \text{rng.random()}.

Since $0 \le u < 1$ , faces are 1..6 with equal probability.

Worked example

If the first four rng.random() values were 0.10, 0.95, 0.50, 0.33, the faces are 1, 6, 4, 2; the first two-dice roll sums to 7, the second to 6; running means 7.0, 6.5.

Common mistakes

Using the global random.seed, which other code can disturb.
int(u * 6) without + 1 (faces 0..5) or round(u * 6) (faces 0..6 with uneven odds).
Expecting the running mean to equal exactly 7.

Why this matters

Monte Carlo simulation estimates quantities that are hard to compute exactly — expected returns in RL (RL-11), bootstrap confidence intervals (ST-13), dropout's expected behaviour.

References

random — Generate pseudo-random numbers — Python Software Foundation (S004)
Think Stats (3rd edition) — Allen B. Downey (S010)