The host opened a goat door, should you switch?

Question

Behind three closed doors are one car and two goats. You pick Door 1. The host, who knows what is behind every door, opens one of the other doors to reveal a goat (Door 3), and offers to let you switch to the remaining closed door (Door 2).

If you switch to Door 2, what is the probability you win the car?

Accepted Answer

When you first picked Door 1, the car was behind it with probability 1/3, and behind one of the other two doors with probability 2/3. The host never opens the car and never opens your door, so his action gives away information. Consider the two cases for your original pick: - Your door has the car (chance 1/3): switching loses. - Your door has a goat (chance 2/3): the host is forced to reveal the other goat, so the remaining door hides the car, and switching wins. P(	ext{win by switching}) = \frac{2}{3} \approx 0.667. The 2/3 chance that the car was "somewhere else" gets funneled entirely on…

The host opened a goat door, should you switch?

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