What do you do?

Question

You are presented with 3 doors from which to choose. Behind one of them, there is a prize, the other 2 are empty. You select a door but before being shown what's there, one fo the other doors is opened and shown to be empty. You are given the choice to change your pick to the remaining unopened door. Do you change? What are the odds of winning the prize and why.

This puzzle has caused raging debate and people generally don't like the answer.

Answer 1

You had a 1/3 chance originally.

There was a 2/3 chance it was behind one of the other doors.

These odds cannot change.

You are shown one of the other doors. It is not behind it. Therefore the odds it is behind the remaining door are 2/3.

You are twice as likely to win by switching.

Answer 2

Some people would reason as follows: When you choose a door, the chances of the prize being behind one of the other two doors is 2:3. But once one of those other two doors are opened to not reveal the prize, then the remaining (unopened, unchosen) door maintains the 2:3 chance, and thus you have a greater chance of finding the prize if you switch your decision.

This is absolutely incorrect.

The revelation of one door alters the probability of the remaining two doors, because you now know that the revealed door is empty. The probably then reverts to 1:2 chances for each of the remaining doors.

The fallicy can easily be shown by considering if there were only two doors to begin with. Beforehand, the probability of choosing the correct door is 1:2. But if the 2nd door were opened and shown to be empty, then the probability for the first door does NOT remain 1:2. The revelation of the empty second door changes the probability of your original choice to be exactly 1:1.

(The infamous counterintuitive solution is achieved when the host who shows the empty door knows which of the two doors is empty. Then the host's knowledge is also a part of the equation, not just your own. If this were the case, then it is always beneficial to switch your choice. Your original question, however, does not state if the empty door was opened randomly or by an intelligent selection from the host.)

Answer 3

You change doors. If the door was opened randomly (in other words, there was a chance that there was something behind it) then it wouldn't matter, but the door was chosen smartly by whomever opened it.

If you don't believe it, here is an experiment.

Take three playing cards A, K, and Q. Have someone mix them up. Pick a card, and have the other person flip over a card that isn't a Q. Then tally the number of times that you should have switched (you are trying to get the Q) to the number of times you shouldn't. Try it, in the end you should see a 2:3 trend to switching. It has been explained correctly mathematically, if you don't believe it, try it.

Answer 4

Since there is 2 remaining doors which may or may not have the prize behind it each door has a 1in 2 chance of hiding the prize. Changing doors does not change your odds in any way. Choosing a door could essentially be compared to flipping a coin. Each side of the coin has an equal opportunity to come up and each door has an equal opportunity of hiding the prize.

Answer 5

Eulercrosser is absolutely correct. You should change doors. The door you originally chose still has a 33% chance, while the other door now has a 50% chance of being the "winning" door. This classic math problem even fooled Marilyn Vos Savant for a while, until she figured it out.

It's counterintuitive, but if our intuition were worth a nickel it would be a happier planet, wouldn't it?

Here is the Wiki, that makes it clear as mud why changing is the best strategy!

http://en.wikipedia.org/wiki/Monty_Hall_problem

Answer 6

The question, as stated, is actually too vague. It says: "One of the other doors is opened and shown to be empty."

Was the door opened by someone who knew what was behind all three doors? If so, you should definitely switch doors to the remaining one that you didn't pick.

But if it was chosen randomly, and just happened to be empty, then switching neither helps nor hinders you.

Therefore, you should switch, because it might help and it can't hurt. :)

Answer 7

With 3 doors you had a 33.3% chance of guessing the correct one (1 out of 3). Now you have 2 doors, so you have a 50% chance of guessing the correct one (1 out of 2). I would probably just stick with the door I originally guessed.

Answer 8

stay with the door u have unless, that is the one that was opened with no prize behind it. chances are 1/2

Answer 9

This very question was on an episode of NUMB3RS earlier this year. You've a 2/3 chance of winning by changing your pick.