What you see depends not only on the world, but also on the rule that made particular information visible to you.
Monty Hall is a probability puzzle, but its deeper lesson concerns the observation protocol. An opened door is informative because the host knows where the prize is and chooses which door to open according to a state-dependent rule.
What you see depends not only on the world, but also on the rule that made particular information visible to you.
The prize is equally likely to be behind any of the three doors. The host knows its location, always opens another door with a goat, and always offers a switch. Under this standard protocol, staying wins with probability 1/3 and switching with probability 2/3.
Under the stated protocol, always switching wins with probability 2/3 overall. A different choice between two available goat doors does not change that strategy probability, but can change the probability after one particular door is revealed. Two remaining doors are therefore not automatically equally likely; a conditional 50/50 result is possible under some host rules.
Distinguish the probability of a fixed strategy before the reveal from the probability conditional on the particular door revealed. The second question also requires the host’s rule when two goat doors are available. With an equally random choice, the conditional switching probability remains 2/3; under a different rule it can differ.
AI transfer to investigate: AI filtering may create the same structural error when users see only the surviving options and the state-dependent selection rule remains unknown.
Proposed test, not yet conducted: Compare displays that hide the host protocol, describe it or show it as a decision tree. Hold prior probabilities and the full host rule constant within each comparison. Test strategy probabilities and probabilities after a particular reveal separately. Score 50/50 as an error only when the specified protocol and question exclude that answer. If the protocol information given does not determine a unique answer, correctly identifying the missing information counts as correct.
Provenance also means: who or what decided that you would see precisely this option?
The 1/3–2/3 result holds for the strategies under equal prior probabilities and a host who always opens another goat door and offers a switch. A different selection rule can give different probabilities.
Gill (2011), The Monty Hall problem is not a probability puzzle, publication reference.
Author manuscript dated 12 November 2010, §2.1, propositions 1–3 and proofs, pp. 6–8.
Gill’s author manuscript treats strategy probabilities and probabilities after a particular reveal as separate questions. The version and passages used are identified above. The application to AI selection has not been tested.
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