Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? What do you think and why?

Question

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? What do you think and why?

Answer 1

Ah, the famous Monty Hall problem.

If you google "Monty Hall problem" , you will get many explanations often in
Youtube form, clicking on one will lead to others.
Here is just one of them:
https://www.youtube.com/watch?v=UgKrQ2ywVfs

Answer 2

I've got the answer now

Answer 3

This question is known as the Monty Hall problem. To analyze it, let's consider the probabilities involved.

Initially, when you pick a door, there is a 1/3 probability that you have chosen the car and a 2/3 probability that you have chosen a goat.

When the host opens another door revealing a goat, this provides you with new information. Since the host knows what's behind each door and intentionally reveals a goat, the probability that the car is behind the remaining unopened door has increased.

Now, let's consider the two scenarios:

Scenario 1: You stick with your initial choice:
If you stick with your initial choice, the probability of winning the car remains 1/3.

Scenario 2: You switch your choice:
If you switch your choice to the other remaining door, the probability of winning the car increases to 2/3.

To explain why switching is advantageous, let's consider the possible combinations of events:

1. If you initially picked the car:
- In this case, the host could open either of the other two doors with a goat. If you switch, you will switch from the car to a goat.

2. If you initially picked a goat:
- In this case, the host is guaranteed to reveal the other goat, leaving one door with a car and the other with a goat. If you switch, you will switch from a goat to the car.

So, in Scenario 2, switching your choice doubles your chances of winning the car from 1/3 to 2/3. Therefore, it is to your advantage to switch your choice.