Home > Riddles > The Monty Hall Problem

The Monty Hall Problem

Chance and uncertaintyLevel 4 · Advanced · ●●●●○

You are in a contest with three closed doors. Behind one there is a car (prize), behind the other two there are goats.

You choose a door (let's say door 1). The presenter, who KNOWS where the car is, opens one of the other two doors showing a goat (let's say door 3).

Now it offers you: 'Do you want to change your choice to door 2, or keep door 1?'. What should you do to maximize your probability of winning the car, and what is that probability?

Solve this riddle

Hints

Show hints

Key idea: Your initial choice “stays” with 1/3 probability. The presenter does not create new probability: he only reveals information, and that is why 2/3 is concentrated on the alternative door.
Step 1 (before the presenter does anything): Your chosen door: 1/3.
CHANGE. Probability of winning: 2/3.

Solution

Show full solution

Answer: CHANGE. Probability of winning: $2/3$.
Analysis:
Key idea: your initial choice “stays” with $1/3$ probability. The presenter does not create new probability: he only reveals information, and that is why the $2/3$ focuses on the alternative door.
Step 1 (before the presenter does anything):

Your chosen door: $1/3$
The other two doors together: $2/3$

Step 2 (the important thing about the presenter):

The presenter knows where the car is
Always open a door with a goat
Therefore, its opening is not a random event: it “redistributes” the $2/3$ towards the only non-chosen door that remains closed

Conclusion:

If you stay, you win with probability $1/3$.
If you switch, you win with probability $2/3$.

Answer: Switching gives $2/3$ probability of winning.

Related riddles

Keep practicing

If you enjoyed this one, try more pure-logic riddles, explore this theme, browse the full archive, or read the riddle-solving guide.

← Previous: The twenty five horses · Next: Cheryl's Enigma (Birthday) →