Estimate pi, integrals, and probabilities using Monte Carlo simulation. Watch random points accumulate and convergence improve.
Random trials replace exact algebra in this estimator. It approximates pi from geometric hits, definite integrals from sampled function values, and probabilities from the proportion of trials satisfying an event. The accumulating-point display shows the estimate changing as the sample grows, making convergence visible rather than presenting only a final number. The usual surprise is that repeating an identical setup can produce a different result. More trials generally reduce sampling noise, but they do not guarantee that any particular decimal place is correct.
In the pi experiment, each random point is classified according to whether it falls inside the displayed circular region. The estimator rescales the observed inside-point proportion using the known relationship between the circle and its surrounding sampling region. The estimate is updated as points accumulate, so its early movement is usually larger and more irregular than its later movement.
For an integral, the estimator samples function values over the entered bounds and combines them into an area estimate. For a probability, it repeats trials and divides the number satisfying the stated event by the total number of trials. In both cases, the displayed result depends on the finite random sample and may approach a stable value unevenly rather than monotonically.
Each run observes a new finite collection of random points. The proportion landing inside the relevant circular region therefore changes, and the derived pi estimate changes with it. Increasing the number of points usually narrows the run-to-run variation. It cannot force every run to return exactly 3.14159, because the displayed value remains a statistical estimate.
No, not when an exact value, guaranteed precision, or certified error limit is required. Monte Carlo results are useful for approximation, experimentation, and convergence demonstrations. Repeat the simulation and increase the trial count to assess stability. For formal numerical work, compare the result with an analytic solution or a numerical method that supplies a defensible error bound.
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