# Do you choose the other door? [duplicate]

You are on the final round of a TV gameshow, there are 3 doors; A, B and C. Two doors lead to a goat and the other door leads to a sports car.

In each episode the presenter asks the contestant to select a door. He then opens one of the unselected doors to show a goat, the contestant is allowed to change their mind and select the remaining unselected door.

Do you choose the other door or stick with your original choice?

Yes, you do.

The two doors that you don't choose have $2/3$ chance that the sports car is behind that door. The presenter opens a door, but he will never open a door with the car behind it. The door that isn't opened by the presenter has therefor still $2/3$ chance to have the car, while your door has only $1/3$ chance. So switch, and double your chance to win!

(For a better explanation, see Monty Hall problem)

• In order to understand it intuitively, just imagine there are 1,000,002 doors. So it's highly improbable you chose the right door. The presenter opens 1,000,000 doors with goats, which is always possible. Now the correct answer (switch) also "feels" right :-) – Ronald Oct 20 '14 at 10:29
• Derren Brown explains it very well in Svengali, using a very large sample, like ronald said. I'll try to find a transcript – Mac Cooper Oct 20 '14 at 11:28

Yes and No, and yes I know that this has been answered, but I guess it deserves a simpler explanation for people like me for both my Yes and No.

There are three doors, you choose 1, there is a 1/3rd Chance of you being right.

Lets say door 1 has the car and doors 2 and 3 have goats.

Scenario 1: You chose 1

The presenter will open one of the leftover doors (either 2 or 3) with the goat and ask if you would like to switch, if you switch, you will lose

Scenario 2: You chose 2

The presenter will not open door 1 he will open door 3, in this case if you switch you will win

Scenario 3: You chose 3

The presenter will not open door 1 he will open door 2, in this case if you switch you will win

Given that the original probability was 1/3 and after the door opening, the odds of winning with a switch is 2/3, you are better off switching.

This, however is the logical answer, then there is the philosophy, if you had a reason to choose door 1 and you believe that reason has not changed, you should not switch, because in life, being wrong about chance is ok, but not sticking to your decision is a bigger problem :) than losing a sports car

• I like this answer, +1 for the philosophy part! – warspyking Oct 20 '14 at 10:31
• The question is about probability, not philosophy. Therefore the only valid answer is YES. – Mast Oct 20 '14 at 11:51
• The question "Do you choose the other door or stick with your original choice?" so I answered from "my" point of view – skv Oct 20 '14 at 12:04
• "not sticking to your decision is a bigger problem". I partially agree with your philosophy. It's good to stand by your choices, but only as long as conditions remain constant. Just as you should remove your coat once the afternoon sun warms the room, so too should you switch doors once the presenter opens a door. – Kevin Oct 20 '14 at 12:38
• 100% agree Kevin, I will modify the answer to reflect this – skv Oct 20 '14 at 12:41

This is the Monty hall problem.

If the presenter knows which door has the prize and never opens it, then you choose the other door. If he doesn't (and therefore has a probability of opening the door with the prize) it doesn't matter whether you change doors or not.

• Your answer is good, but the useful information it contained has already been provided. – d'alar'cop Oct 20 '14 at 17:24

Yes, you switch. A critical point is that you say

In each episode the presenter asks the contestant to select a door. He then opens one of the unselected doors to show a goat.

This lets us know he is (probably) deliberately choosing a goat door, thus providing you with new information. If he was randomly choosing one of the two unselected doors, seeing a goat would tell you nothing. In that case, you would expect him to reveal a car in $\frac{1}{3}$ of the episodes.