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AnotherKevin comments on Navigating disagreement: How to keep your eye on the evidence - Less Wrong
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The thermometer answer is wrong, you're ignoring that you're on a game show On a game show the producers try to organize things such that few people (or only one person) wins a challenge. As such I would expect all but one thermometer to be in error. Furthermore by watching old episodes of the show I could tell if only one thermometer will be right or if several contestants also succeed at each challenge and therefore either pick the small clump or the lone outlier.
This is a very good point. Since you might be being messed with, you should run every sanity check you can think of. In increasing order of difficulty and also increasing order of value: get the room temperature from all the thermometers; take your own temperature; ask for a drink of ice water and take its temperature. You should also consider the possibility that all of the other contestants are actors with fake thermometers.
It's a metaphor, like the Monty Haul problem. The fact that that's not how game shows really work doesn't matter.