Monty Hall Problem, explained.
The Monty Hall problem is a conditional probability puzzle: switching doors wins with probability two-thirds when an informed host always reveals a losing unchosen door and offers a switch.
Why it happens
Your first choice has a one-third chance of being right. The other two doors collectively have two-thirds. The host's constrained reveal concentrates that remaining chance on the one unchosen door left closed; it does not reset the original choice.
Your first choice wins one time in three. An informed host always opens an unchosen goat door and always offers a switch. Switching wins precisely when your first choice was wrong.
Read the result
Compare many stay and switch trials. The advantage depends on the host knowing the prize location and following the stated reveal rule. An uninformed or selective host creates a different problem.
A worked example
Imagine 300 independent rounds
About 100 initial choices are correct and about 200 are wrong in expectation.
Staying wins in the first group. Switching wins in the second because the informed host removes the other losing door.
Switching therefore wins about 200 rounds, although actual finite counts fluctuate.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
The prize is uniform across three doors. If the host has a choice of goat doors, the host chooses randomly. Batch results evaluate staying and switching on the same games; their wins sum to the number of games.
Where this idea is useful
A practical use
When interpreting an interview shortlist or a revealed clue, ask how the information was selected. A deliberate reveal and an accidental observation can support different inferences.
A common misconception
“Two closed doors means fifty-fifty.”
The route by which the host removed a door carries information. Counting doors without accounting for that process loses it.
What this explanation leaves out
- The two-thirds switching result depends on this host policy. A host who sometimes reveals the prize or selectively offers a switch changes the problem.
Why doesn't the host's reveal change my original chance?
Under the standard rule, the host can reveal a losing door whether your first choice is right or wrong. Your choice still has its original one-third chance; the alternative inherits the two-thirds chance that you initially missed.
Which host rule would make the usual switching argument stop applying?
Further reading
Explore the original research or the teaching reference behind this experiment.