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Cyan comments on The Monty Maul Problem - Less Wrong
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The Monty Crawl variation is interesting. If you have to pick an unchanging strategy for a large number of trials, always switching wins two thirds of the time. But in any particular trial, after Monty opens a door you either know with certainty that switching wins (one third of the time), or switching and sticking are both 50:50 (two thirds of the time).