Problem — Simulating Monty Hall with a die

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Simulate the Monty Hall problem with a $6$-sided die: doors $1$, $2$, $3$; use three die outcomes for the position of the prize ($1$–$2$ = door $1$, $3$–$4$ = door $2$, $5$–$6$ = door $3$). Play at least $30$ trials and compare the winning frequency of the «switch» and «stay» strategies with the theory.

Solution

Set-up. Fix a strategy (for example «always switch»). For each trial: roll the die to place the prize, choose an initial door, the host opens a door with the goat among the two remaining, then apply the strategy. Record win/loss. The theoretical prediction is obtained with Bayes’ theorem / counting the cases: by switching you win with probability $2/3$, by staying with probability $1/3$. After $30$ trials the experimental frequency of winning «by switching» should approach $2/3 \approx 20$ wins out of $30$. Discuss the deviations as an effect of the finite sample.

Links

Topics: Probabilita
Concepts: Paradosso di monty hall · Teorema di bayes
Methods: Bayes
Skills: Calcolo probabilita · Ragionare per casi
Exercise type: Problema probabilita