There are 3 doors. 1 leads to success the other 2 to failure. You are invited to pick one door.?

Question

You choose and just before you open it you are stopped by the supervisor who knows the answer. He says “FYI it is not this door” and opens one of the doors you haven’t picked. He then invites you to change your mind. Should you change you mind or not? Why?

Answer 1

You should change doors. This is the infamous Monty Hall problem and its explanation relies on the concept of conditional probability. This medium makes it hard to explain with any clarity but I shall be happy to email anyone an explanation if they want one. You might Google 'Monty Hall'. In short, the prob of winning without switching is one third and with switching is two thirds.

Answer 2

Making the assumption that the supervisor would've opened a wrong door for you no matter what you picked, you should change your mind. You have a 2 in 3 chance of getting the success if you switch and 1 in 3 chance if you don't.

This is a famous problem called the "Monty Hall" problem after the game show host, and is admittedly hard to wrap one's mind around. But let's say that you choose door 1. There's a 1 in 3 chance that 1 is the success door and the supervisor can open either door 2 or door 3 and you'd lose by switching. But there's a 2 in 3 chance that 1 is a failure door. If door 2 is the right one then, the supervisor opens door 3 and if door 3 is the right one then the supervisor opens door 2. Without knowing any better you're better off switching.

One way to think about it is if you had 100 doors, with only one leading to success and if you selected one, then the supervisor showed you 98 "failure" doors and invited you to choose between the door you selected and the door he didn't open. It should be more obvious that it's a good idea to switch then. Or you can try simulating it. Have a friend randomly choose a number from 1-3. You make a guess then the friend chooses a wrong number that you didn't pick, then you decide to switch. Do this a bunch of times and see how often switching pays off. Many classroom and computer simulations that were done many times have shown that switching works about 2/3rds of the time.

Answer 3

To satisfy both answers given so far, the key is whether or not you choose a door in the first place.

If you don't choose a door in the first place and your supervisor opens one of the doors leading to failure, and then you choose a door, the chances of that door leading to success are fifty-fifty. With this scenario the switch doesn't alter your chances.

But if you do choose a door in the first place, then it goes from basic probability to conditional probabilty. There is a 2 in 3 chance that the door you chose will lead to failure, and if you have chosen a door that leads to failure then your supervisor is forced to open the other door that leads to failure. There are 3 possibilities and in 2 out of these 3 possiblilities the supervisor has no choice as to which door he opens.

The sample space is
S F F
F S F
F F S

Consider choosing any Door.

If the Door leads to Success the Supervisor can show you any of the other 2 Doors. You switch, you get a Failure Door.

If the Door leads to Failure the Supervisor is forced to show you the other Failure door. You switch, you get a Success Door

So switching you have a 2/3 chance of Success whilst sticking you have a 1/3 chance of Success.

Answer 4

You should switch. S=Success, F= Failure. Since you are picking a door at random, lets say you pick the door#1 as it does not matter. The 3 possible combination of doors is as follows:

1 2 3
---------------
a) S F F
b) F S F
c) F F S

Scenario a): Supervisor can choose either door #2 or #3, if you switched you would have lost.
Scenario b): Supervisor has to choose failure door #3, if you switched you won!
Scenario c): Supervisor has to choose failure door #2, if you switch you won!

1/3 chance of winning if you stay with original choice, 2/3rds if you switched.

Answer 5

Yes, you should change your mind. This only works because the supervisor KNOWS which door is which.

To begin with, each door has a 1 in 3 chance of being the right door. You picked a door with a 1 in 3 chance. When the super opens one of the other doors, the remaining door has a 50:50 chance of being the right door. So you should switch.

MANINDER: Well argued, but unfortunately that's not right!

Try it yourself on paper. It's counterintuitive, but you really do improve your chances!

Better yet (and for a clear explanation) you can try it here if you have no paper:

Answer 6

It would depend on why this supervisors motives are. Does it benefit him to have me fail or succeed? Also, if he was going to show me the right way, why didn't he just do it in the first place. By virtue of him letting me start to choose and then stopping me, I would have to assume that I picked the correct door for success and the supervisor will not benefit from my choice, he may even be hurt by it and so he is now trying to get me to choose the wrong door. So, I would stick with my own choice, even if it was failure, at least I would know that I made the choice on my own and that I am responsible for the choices I make, not someone else.

Answer 7

It's the Monty Hall problem

There are three scenarios when you pick a door, the probability of each scenario is 1/3

1) You pick the success
2) You pick failure 1
3) You pick failure 2

If one of the remaining "failure" doors is revealed then if you switch you will get the following outcomes

1) Failure
2) Success
3) Success

If you don't switch you get the following outcomes

1) Success
2) Failure
3) Failure

If you switch you therefore have a 2/3 chance of winning. If you don't switch there is only a 1/3 chance you selected the correct door in the first place.

Answer 8

Do not change your mind. Go with your gut feeling. That door might not be the right door for the supervisor, it doesn't mean it's not the right door for you personally. Even a catastrophe can open the door to a wonderful new beginning.

Look at the Katrina hurricane and all those families who got BRAND NEW homes from various sources including Oprah. I'm sure they would not have consciously picked that door, but it was the one for them.

Answer 9

Change your mind.

Bayes Theorem:

Consider the position when door 3 has been chosen and no door has been opened; the probabilities P(C1) that success is behind door 1, P(C2) (success behind door 2), and P(C3) (success behind door 3), are all plainly 1/3. The probability that the supervisor will open door 1, P(O1), is 1/2:

P(01) = P (C1) x P (01/C1) + P (C2) x P (01/C2) + P (C3) x P (01/C3)

= 1/3 x 0 + 1/3 x 1 + 1/3 x 1/2 = 1/2

When success is behind door 1, the supervisor will never open door 1.
When success is behind door 2, the supervisor will certainly open door 1.
When success is behind door 3, he will open door 1 or door 2 with probabilities 1/2.

Hence the probability that success is behind door 2 given that the supervisor opens door 1 is:

P (C2/01) = [P(01/C2) x P(C2)] / P(01) = [1 x 1/3] / 1/2 = 2/3

In the same way, you may find P(C1|O2) that is also 2/3.

Try a simulator and test it for yourself:

http://www.userpages.de/monty_hall_problem/

http://www.grand-illusions.com/monty2.htm

http://alterlife.org/2006/04/16/a-javascript-simulator-for-the-monty-hall-problem/

Answer 10

Well, this is an old problem related to probability.

You may already be knowing that there are two schools of thought on this. Hence I am going to give logic for the one I follow.

Its simple. After the supervisor has told you about the door which ain't correct, the problem changes with the introduction of this new information.

Now the problem is very simple. Two doors, one good and the other bad. Therefore there should be no advantage for you even if you change (as the other school of thought followers argue that one gets an advantage f he changes his choice now).

For me its the same.. .with 50% probability for either door!!!

Answer 11

But if the supervisor was going to tell me the right one anyway why bother choosing. If the supervisor is definitely truthful and not trying to trick me I would change my mind. Better to have someone who knows better lead you.

But if the supervisor is a liar I would proceed with my first decision at my own risk. In life we have got to make choices.