You are on a game show with four doors. Behind one door is a car, and behind the other three are goats. You choose a door, and the host, who knows what's behind each door, opens one of the remaining three doors to reveal a goat. You're then given the option to switch. What is the probability of winning if you switch?
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The probability of winning if you switch is higher than if you stick with your initial choice.
Initially, when you choose one of the four doors, the probability of choosing the car is 1/4 (25%) while the probability of choosing a goat is 3/4 (75%).
When the host opens one of the remaining three doors to reveal a goat, the probabilities change. The car is still behind one of the doors, but the probability of your initial choice being the car is still 1/4 (25%), and the probability of your initial choice being a goat is now 0. The host's choice, which was initially a mix of goats and the car, becomes a certainty of being a goat.
With one goat door revealed, 3 doors remain - your initial choice, one unopened door, and the goat door revealed by the host. The car is still behind one of these doors. The probabilities for each door change. The probability of your initial choice being the car remains at 1/4 (25%), the probability of the unopened door being the car is now 3/4 (75%), and the probability of the revealed goat door being the car remains at 0.
When given the option to switch, you should switch because the probability of the unopened door being the car is higher (3/4 or 75%) compared to the probability of your initial choice being the car (1/4 or 25%).
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