Monty Hall: A Short Explanation
The Monty Hall problem, based on the game show Let’s Make A Deal, still stumps a lot of people.
I arrived at the following short explanation over lunch one day with coworkers.
First, a brief explanation of the rules and the question.
- The host, Monty Hall, shows a contestant three closed doors, numbered 1, 2 and 3. Behind one door is a valuable prize, and behind the other two are worthless prizes, usually a goat or a much less valuable prize.
- The contestant first choose a door, at random.
- Then Monty opens one of the doors the contestant did not choose, revealing a goat or very low value prize. Important: He never reveals the prize at this point.
- Monty then tries to persuade the contestant to change their mind, offering cash in some case.
The question:
Should you, as the contestant, stay with your choice, or switch your choice to the other closed door?
The simple explanation:
Two thirds of the time:
(You choose one wrong door and the host shows you the other wrong door).
It really is that simple. Two thirds of the time: you chose one of the wrong doors, the host shows you the other wrong door, and you should switch. I add the parentheses here to emphasize that the two clauses in parentheses happen together.
You also can work this out by making a table of all the possible games and choices. There aren’t that many.
It’s interesting that a lot of effort and (virtual) ink is still used in concocting detailed explanations, often with some fairly advanced probability theory!