Well my explanation for this problem has come under a lot of criticism. I would like to share it.

It is my conjecture that if these were 3 boxes and one of them had lets say $1000 dollars in it while the other two were empty then we have the same problem at hand.

I say that if Monty did not open the empty box and asked the contestant whether they would like to swap their one box for his two boxes then the result would be no different.

The contestant would agree that Monty has twice the likelihood of having the prize, since he has two boxes. And hence it would be more attractive to swap.

I then get the person I am trying to explain the solution to to admit that indeed one of Monty's boxes will always be empty. So whether he leans over and opens the box after he handed the two boxes to the contestant (after the swap), or whether he opens it first and then asks you to swap the unopened box, makes no difference. Monty is twice as likely to have the prize because he has two boxes, the contestant has one box and one of Monty's boxes is always empty.

I have a brief comment about the Monty Hall problem. I just today found your applet which I believe contains the clearest explanation of it, one I have used for years.

I want to restate it more explicitly, and free from the applet:

Original problem:

There are three doors with a prize (a car) behind one of them and a poster of an undesirable goat behind each of the other two. You are asked to select a door which you do. Monte then opens one of the two doors you did not select to reveal a Goat poster and asks whether you now want to switch to the third door (which you did not select and which Monty did not open).

You get the prize if it's behind the door you open, otherwise you get nothing. Should you switch?

Equivalent formulation:

There are three doors with a prize (a car) behind one of them (and a poster of an undesirable goat behind the other two.) You are asked to select a door which you do. Without opening any of the doors, Monty asks you whether you want to stay with the door you selected, or switch and open both of the other two doors.

You get the prize if it's behind one of the doors you open, otherwise you get nothing. Should you switch?

It is easy to see that the reformulation is equivalent to the original problem and it is also easy to see that opening the door you chose gives you a 1/3 chance of winning and opening the other two doors gives you a 2/3 chance of winning. (Sorites reasoning.)

I also use a Lemon instead of a goat because I am a goat lover.

Best wishes, Sandy Lemberg