Conditional-probability problem
Monty Hall Problem
With three doors, a prize placed uniformly, and a host who always reveals a goat behind an unchosen door, switching wins with probability two thirds.
P(win by switching) = (n - 1) / n for an n-door informed-host variant
For the classic three-door protocol the probability is 2/3. If the host does not know, may reveal the prize, or chooses doors by another rule, the answer changes.
Each cycle marks the initial choice, the hidden prize, the host's constrained reveals, and the one remaining alternative.
(%)
The animation runs automatically, pauses on the conclusion, and then repeats. The main control changes the scenario rather than scrubbing the timeline.
- CHANGE
- Number of doors
- WATCH
- switching win probability
- MEANING
- Each cycle marks the initial choice, the hidden prize, the host's constrained reveals, and the one remaining alternative.
The host removes doors without removing the initial choice's probability.
Repeated trials split into "first choice right" and "first choice wrong"; switching wins the entire second branch.
What it actually says
The initial choice has probability 1/3 of being correct and the unchosen set has probability 2/3. The informed host concentrates that set's probability on the only alternative left closed.
The puzzle feels counterintuitive because the host's action is mistaken for random missing information. It is a selection event constrained by knowledge of the prize.
"A useful law compresses a pattern. It does not erase the conditions that make the pattern true."
How the idea developed
The modern form emerged through observation, argument, and later refinement. The timeline separates the first insight from the version now used in textbooks and practice.[1]
Steve Selvin presents the problem in The American Statistician.
A popular column triggers widespread debate and simulation.
The problem teaches conditional probability and information protocols.
How the pattern works
The relation becomes useful only when its mechanism, measurement process, and operating range are visible.
One chosen door versus all unchosen doors.
The host avoids both the prize and the selected door.
The unchosen-set mass moves to the surviving alternative.
For the classic three-door protocol the probability is 2/3. If the host does not know, may reveal the prize, or chooses doors by another rule, the answer changes.
Where it earns its keep
Applications are strongest when the law changes a decision, measurement, model, or experiment rather than merely providing an analogy.
Enumerate complete cases
ApplicationList prize location, host action, and strategy outcome.
State the protocol first.
Model selection mechanisms
ApplicationObserved absence can be informative.
Ask who chose what to reveal.
Where it stops working
Variants with a forgetful, adversarial, or probabilistic host have different posterior probabilities.
"After one reveal the doors are symmetric"
Better: The initial choice and host-selected survivor have different histories."Switching guarantees a win"
Better: It raises the probability from 1/3 to 2/3.Sources and further reading
Original publications and serious secondary scholarship are prioritized over summaries.
- Selvin - A Problem in ProbabilityOriginal published formulation.https://doi.org/10.1080/00031305.1975.10479121
- Gillman - The Car and the GoatsFormal analysis of variants.https://doi.org/10.1080/00031305.1992.10475842
- MIT OpenCourseWare - Conditional ProbabilityAuthoritative teaching context.https://ocw.mit.edu/courses/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/pages/unit-i/lecture-3/