Monty Hall paradox

You are on a game show: three closed doors. Behind one there is a car, behind the other two a goat. You choose door $1$. The host, who knows what is behind each door, opens door $3$ and shows a goat. He asks you: “do you want to keep door $1$ or switch to door $2$?”.

Intuition says: two closed doors, a $50$–$50$ choice, it does not matter whether you switch. Wrong. Switching doubles the probability of winning ($2/3$ against $1/3$).

Property — Explanation via Bayes

Let $A_i =$ “the car is behind door $i$”, $B =$ “the host opens door $3$”. By symmetry $P(A_1)=P(A_2)=P(A_3)=1/3$. Knowing the host’s choices:

• $P(B\mid A_1) = 1/2$ (car behind $1$: the host opens $2$ or $3$ at random, $50\%$);
• $P(B\mid A_2) = 1$ (car behind $2$: the host must open door $3$, he cannot open door $2$);
• $P(B\mid A_3) = 0$ (he would not open the car’s door).

Total probability: $P(B) = \tfrac13\cdot\tfrac12 + \tfrac13\cdot 1 + \tfrac13\cdot 0 = 1/2$. We apply Bayes: $P(A_1\mid B) = \frac{P(B\mid A_1)P(A_1)}{P(B)} = \frac{(1/2)(1/3)}{1/2} = \frac{1}{3}, \qquad P(A_2\mid B) = \frac{1\cdot(1/3)}{1/2} = \frac{2}{3}.$

Observation — Why intuition is wrong

The host does not choose at random: he has information that we do not have. His conditional choice “transfers” the probability $1/3$ of the discarded door onto door $2$, which becomes $2/3$. Door $1$ stays fixed at $1/3$ because our initial choice is set before the host acts, so it receives no update. It is the same mechanism as diagnostic tests: it is not enough to count the closed doors, one must know how the current state was reached. The classic narrative of the paradox (with the popular debate sparked by Marilyn vos Savant’s answer in the 1990s) is told by Dunham (Ch. 11).

Example — Simulation with $99$ doors

Amplified version: $100$ doors, $1$ car, $99$ goats. You choose door $1$; the host opens $98$ doors showing $98$ goats, leaving you to choose between door $1$ and a mystery door. The same logic says: $P(\text{win if you switch}) = 99/100$. Made explicit in this form, the paradox disappears.

Links

Topics: Probabilita
Concepts: Paradosso di monty hall · Probabilita condizionata · Teorema di bayes
Methods: Bayes · Prob condizionata
Skills: Calcolo probabilita · Ragionare per casi