Monty Hall — Why host policy matters

Small Monte Carlo simulation that shows two versions of the Monty Hall game:

1. Standard host (textbook Monty Hall): the host always opens a door with a goat and then offers a switch. In this case, the host’s behaviour is informative (he avoids the car). Switching wins with probability 2/3; staying wins with 1/3.

2. Random host (conditioned on a goat reveal): the host randomly opens one of the other two doors. In this case, the goat reveal happened only by chance. Switching and staying both win with probability 1/2.

How we obtained the information (host policy) determines the correct inference.

Quick start

Just run monty_hall_demo.

The script plots the running (cumulative) success rates.

Note:

By Bayes' theorem, the conditional probability formulas are the same, but since the randomized host is not guaranteed to reveal a goat $(\mathrm{P}(\text{goat reveal} \mid \text{car not behind your door})$ and $\mathrm{P}(\text{goat reveal})$ are not $1)$ the probabilities shift:

1) In case of the standard host:

$$\begin{aligned} &\mathrm{P}(\text{goat reveal} \mid \text{car not behind your door}) = 1 \\\ &\mathrm{P}(\text{car not behind your door}) = \frac{2}{3} \\\ &\mathrm{P}(\text{goat reveal}) = 1 \\\ \\\ &\mathrm{P}(\text{car not behind your door} \mid \text{goat reveal}) = \frac{\mathrm{P}(\text{goat reveal} \mid \text{car not behind your door}) \cdot \mathrm{P}(\text{car not behind your door})}{\mathrm{P}(\text{goat reveal})} = \frac{1 \cdot (2/3)}{1} = \frac{2}{3} \end{aligned}$$

2) In case of the random host:

$$\begin{aligned} &\mathrm{P}(\text{goat reveal} \mid \text{car not behind your door}) = \frac{1}{2} \\\ &\mathrm{P}(\text{car not behind your door}) = \frac{2}{3} \\\ &\mathrm{P}(\text{goat reveal}) = \frac{2}{3} \\\ \\\ &\mathrm{P}(\text{car not behind your door} \mid \text{goat reveal}) = \frac{\mathrm{P}(\text{goat reveal} \mid \text{car not behind your door}) \cdot \mathrm{P}(\text{car not behind your door})}{\mathrm{P}(\text{goat reveal})} = \frac{(1/2) \cdot (2/3)}{(2/3)} = \frac{1}{2} \end{aligned}$$