You should always switch in the "Monty Hall Problem" because it doubles your chances of winning from 1/3 to 2/3; your initial pick has a 1/3 chance of being right, so the other two doors combined hold a 2/3 chance, and the host's action of revealing a goat concentrates that 2/3 probability onto the single remaining unopened door, making switching the statistically superior strategy.
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