Switching doors after the host reveals a goat
Question
On a game show there are three closed doors: one hides a car and the other two hide goats. You pick a door at random. The host, who knows where the car is, then opens one of the other two doors to reveal a goat, and invites you to stay with your door or switch to the remaining closed one. If you switch, what is your probability of winning the car?
Short answer
. Switching loses only if your first pick was the car, which happens with probability . Whenever you started on a goat, the host's reveal leaves the car behind the other closed door, so switching wins with probability .
Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.
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Open the worked solutionLessons behind this question
- Conditional Probability, Total Probability, and Bayes' Rule
- Counting and Combinatorics
- Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating
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