Estimate Pi with a Square Cardboard and a Fair Die to 5-6 Decimal Places

You are given a square piece of cardboard and a fair six-sided die whose faces 1 to 6 are equally likely. Design an experiment that estimates π empirically with these items. Then explain how you would get the estimate accurate to 5–6 decimal places. ### Constraints and Clarifications - Assume you may draw on the cardboard and measure it with a pen and a ruler; nothing else is provided. - The question does not define "accurate to 5–6 decimal places" further. State the error tolerance and the confidence level you are targeting. ### Clarifying Questions - Is the die meant to be thrown onto the cardboard, or only rolled to produce random numbers? - Must the experiment be carried out by hand, or may the die's role be simulated on a computer once the design is fixed? - Does "5–6 decimal places" mean an error below $h = 5 \times 10^{-6}$ to $h = 5 \times 10^{-7}$, and at what confidence? ### Part 1 — Design the experiment Describe the procedure: what you draw on the cardboard, how the die produces each trial, what you record, and the formula that turns the record into an estimate of π. Explain why your estimate converges to π and not to some nearby number. ```hint Turn π into a probability Look for an event whose probability involves π and that you can decide trial by trial using only the cardboard and the die. ``` ```hint Six outcomes are coarse One roll gives only six possible values. Ask what your estimator converges to if each coordinate comes from a single roll, and how additional rolls change that. ``` #### What This Part Should Cover - A concrete procedure: the geometry drawn, how die rolls map to a trial, and what counts as a success - The estimator and why its expected value is π, or how close to π it is - Sources of bias in the design (coarse random numbers, physical throws that are not uniform) and how the design removes them - The cost of one trial in die rolls ### Part 2 — Reaching 5–6 decimal places How many trials does your design need for 5 and for 6 correct decimal places? Is that feasible, and what would you change to get there? ```hint Error shrinks slowly Work out the standard deviation of a single trial, then how the error of the average scales with the number of trials. ``` ```hint Not all error is random More trials only remove noise. List what else could make the answer wrong by more than one part in a million. ``` #### What This Part Should Cover - The variance of the estimator and the number of trials implied by the target, with the arithmetic - Feasibility in die rolls and time - Ways to need fewer trials: variance reduction, stratified or quasi-random sampling, automation - Systematic errors that more trials cannot fix: die bias, discretization, physical measurement ### What a Strong Answer Covers - Links π to a probability or an integral and builds an estimator whose bias is zero or controlled - Quantifies the statistical error and turns the precision target into a trial count - Recognizes how slowly plain Monte Carlo converges and responds with a better method rather than more rolling - Separates random error from systematic error and reports the result with a confidence interval ### Follow-up Questions - On average, how many die rolls does one trial of your design consume, and how could you lower the total number of rolls without introducing bias? - Suppose the die is slightly unfair. How does that bias your estimate of π, and how would you detect and correct it? - If you could choose an integral to estimate instead of an area, which one would you pick to need fewer trials for the same precision, and why? - How would you decide when to stop rolling, and how would you report the result?