Probability is a branch of mathematics that most people have an instinctive feel for. Unfortunately, our instinctive feel is often wrong. One of the best examples where common sense is at odds with Probability Theory is known as 'The Monty Hall Problem'. When the mathematical solution is given, people are convinced it is wrong. Mathematicians have almost come to blows in discussing this problem.
The problem is inspired by an American game show, Let's make a Deal which ran on NBC throughout the 1960s and 1970s. The host, Monty Hall, would often give contestants a chance to change their mind. The problem as outlined here is not the way Monty actually behaved on the show, but is interesting nonetheless.
In the Monty Hall Problem, there are three identical doors. Behind one of the doors is a valuable prize, say a dream holiday. Behind the other two doors there is nothing. You are allowed to choose one door. If you choose the right one, you get the prize. But before the door is opened, the host (and let's call him Monty) proceeds to open one of the other two doors and shows you that there is nothing behind it. Now you are given the option of sticking with your original choice or changing your mind. What should you do?
Note that Monty is not picking one of the remaining doors at random; he knows which door hides the prize. Whichever door you choose initially, Monty will always open a door which has nothing behind it. If you pick a door with nothing behind it, Monty will open the other door with nothing behind it. If you have chosen the door which hides the prize, Monty will pick one of the two remaining doors at random, so that he still opens a door with nothing behind it.
The Common Sense Answer
The common sense answer is that since Monty has shown you what's behind one of the three doors, there are only two left. Each of these must be equally likely, so there is no reason to change your choice. It is just as likely as the remaining door to hide the prize.
This answer is of course wrong!
The Correct Answer
The correct answer to this problem is that you should change your choice. You should always change to the door that Monty has not opened.
This is so counter-intuitive that it will have to be explained in a few stages. If you follow each of the stages, you should be convinced by the end.
In Stage One, we will consider a simpler version of the game. After you have chosen your door, you are giving the option of either opening it, or changing your choice and opening both the other doors. What should you do?
It is fairly obvious that you should change your choice and go for opening the other two doors. They are twice as likely to hide the prize as the one you have chosen. There is a 1 in 3 chance that your door hides the prize while there is a 2 in 3 chance that the prize is behind one of the other two doors.
So in this case, there is a good reason to change your choice.
In this version of the game, there is a fourth door, which it is guaranteed does not have anything behind it. After you have made your initial choice, Monty opens this door, revealing nothing. Should you stick with your choice or change to one of the remaining two doors?
It doesn't matter whether you change or not. Monty has not provided any extra information about your door by opening the fourth door. You knew in advance that he would open it and that there would be nothing behind it. So your door still has a 1 in 3 chance of being the prize door. The other two doors also have a 1 in 3 chance each. So there is no advantage in changing.
Now we come to the full Monty Hall problem as stated above. When Monty opens the door, we know in advance that he is going to open a door and that there will be nothing behind it. So, just as in stage 2, his opening the door provides no extra information about the door you have chosen. So the chance of there being a prize behind the door you have chosen is still 1 in 3. It has not changed due to Monty opening the door.
The final stage in understanding the solution is to see that if the chance of the prize being behind your chosen door is 1 in 3, then there is a chance of 2 in 3 of it being behind one of the other two doors. We are back to the same as Stage 1. Monty has already opened one of the doors for us, so we should choose to change our choice and open the other one.
Still Not Convinced
Many people are still unconvinced by this explanation. They must come to terms with it in their own head. Here is one researcher's explanation of the solution in his own words.
Could I humbly (well all right - blatantly) recommend my 'explanation', viz. that Monty opening a door is not always acting freely; in those cases where the guest has guest wrong (sorry, couldn't resist that), he is compelled to open the only remaining losing door, pointing out the winning door by default. It only remains to notice that these (forced) occasions outnumber the alternative by two to one.
The solution can be made easier to appreciate by increasing the number of doors. Suppose that there are 50 doors, one of them concealing the prize. You pick one. Now think, is it likely that you have chosen the right one? No, there is a 1 in 50 chance that you have been lucky.
Monty now has the arduous task of opening 48 of the remaining doors, leaving one door closed as he collapses into his gameshow host's chair, panting and out of breath. Monty guarantees that the prize is either behind your door or behind the one remaining door. Should you stick with your choice, or should you change to the one that Monty has tantalisingly left closed?
Still not convinced?
Get a friend to volunteer to help you. The friend can pick a door to hide the prize behind, you make your choice and then the friend can act as Monty. If the friend needs to choose between two doors at random, they can use a coin toss. After 20 or 30 times, you will be convinced that you are better off to change than to stick with your original choice.
The Monty Hall Problem demonstrates that you cannot trust your instincts in even fairly simple situations involving chance. You must sit down and work out the details to check that your intuition is correct.