Probability and simulation

Random sampling can estimate fixed mathematical quantities.

This prototype introduces Monte Carlo simulation through two visual examples: estimating pi and estimating area under a curve.

The aim is to help learners see how probability, geometry and computation can work together.

Monte Carlo lab

Use randomness to estimate something deterministic

Monte Carlo methods use random sampling to estimate quantities that may be difficult to calculate directly. More samples usually improve the estimate, but randomness means the error wiggles.

Estimate

3.14

Exact value

3.1416

Absolute error

0.0016

Hits

471 / 600

Joy in the process

The point is not just to finish. It is to notice, test and return.

These tools are invitations to explore. A good mistake, a surprising pattern or a question you cannot yet answer is part of the work, not a failure of it.

The challenge is deliberate: the site should support thinking, not remove the need for it.

Before changing a setting, pause and predict what you think will happen.

Change one thing at a time. What stayed the same, and what changed?

Try to create a surprising case, a broken case, or a beautiful pattern.

Ask what this connects to outside the page: maps, movement, nature, systems or decisions.

Reset, then try again with a new question in mind.

Future extensions

This can become a broader simulation and numerical methods lab.

Add convergence graphs showing error against number of samples.

Add custom functions for area estimation.

Add confidence intervals and repeated trial summaries.

Add option pricing or risk simulation as an advanced extension.