Imagine the Monty Hall problem had four doors instead of three, and Monty opened one door to reveal a goat after your initial choice, what would be the probability of winning the car by switching? 1/2 1/3 1/4 3/4
Question
Imagine the Monty Hall problem had four doors instead of three, and Monty opened one door to reveal a goat after your initial choice, what would be the probability of winning the car by switching? 1/2 1/3 1/4 3/4
Solution
The Monty Hall problem is a probability puzzle based on a game show scenario. The original problem has three doors, behind one of which is a car and behind the other two are goats. The contestant chooses a door, then the host, who knows what's behind each door, opens one of the other two doors to reveal a goat. The contestant then has the option to stick with their original choice or switch to the other unopened door.
In this modified version of the problem, there are four doors instead of three. The process is the same: you choose a door, Monty opens one of the other doors to reveal a goat, and you can then stick or switch.
If you stick with your original choice, the probability of winning the car is 1/4, because there was a 1 in 4 chance that you chose the correct door at the start.
If you switch, the situation is a bit more complex. When Monty opens a door to reveal a goat, he gives you information about where the car is not. This changes the probabilities of the remaining doors.
Here's how it breaks down:
- There was a 1/4 chance the car was behind your original choice, and a 3/4 chance it was behind one of the other doors.
- When Monty opens a door to reveal a goat, the 3/4 chance that the car was behind one of the other doors gets redistributed among the two remaining unopened doors.
- So, if you switch, there is a 3/4 chance that you will win the car.
So, the answer to the question is 3/4.
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