A prize is hidden behind one of three doors. You choose one door. The host, who knows where the prize is, then opens a different door that certainly hides no prize.

Two unopened doors remain: your original choice and one alternative.

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Should you stay, switch, or does it make no difference?

Assume the host always opens a losing door and always offers the switch.

Hints

Open one at a time

Keep track of what the host knows.

Your first choice is wrong with probability 2/3.

Solution

Best opened after a real attempt

Switch

Your first choice is correct with probability $1/3$ and wrong with probability $2/3$. The host's action does not redistribute that original probability evenly; it reveals information while obeying a rule.

Initial choice	Probability	If you switch
Prize	$1/3$	Lose
No prize	$2/3$	Win

Thus staying wins with probability $1/3$, while switching wins with probability $2/3$.

An equivalent way to make the asymmetry visible is to imagine one hundred doors. Choose one; the informed host opens 98 losing doors. Your single door still carries its original $1/100$ chance, while the one surviving alternative carries the other $99/100$.