7.6 Monte Carlo and randomness

Explanation

Monte Carlo methods use randomness to estimate quantities. A Monte Carlo result should include the random seed, sample size, estimated uncertainty, and a check that the result changes sensibly with more samples.

In Rust, use an RNG crate such as rand. Computers usually generate pseudorandom numbers, not truly random numbers. A pseudorandom number generator follows a deterministic algorithm. The seed selects the starting point of the sequence. If the seed, algorithm, and code are the same, the same sequence can be reproduced.

The seed policy matters. A fixed seed is useful for debugging and reproducibility, but it does not remove sampling error. For a scientific claim, report uncertainty instead of trusting one seed.

Put the main sampling work in a function. A function makes the computation easier to test, rerun with different parameters, and separate from printing, file output, or plotting.

Things to look up

• Monte Carlo method
• Pseudorandom number generator
• Random seed
• Sampling error
• Error bar
• Law of large numbers
• rand

Exercise

Estimate pi in Rust by sampling random points in a square and counting how many fall inside the unit circle.

Run two sample sizes, such as 10_000 and 1_000_000. For each run, record:

• The command you ran.
• The seed or seed policy.
• The sample size.
• The estimate.
• A simple uncertainty estimate or repeated-seed comparison.

Notes for the exercise

• Multiple seeds or uncertainty estimates matter for scientific claims.
• Save the result and metadata before plotting if a plot is made.
• Put the repeated sampling loop in a function.
• Keep plotting, printing, and file output outside the core computation when possible.
• Explain how the error should change as the sample size increases.