The role of probability in solving the Monty Hall problem

[Jenny A]

You are admitting, even bragging about, being ignorant. 

I suggest you study probability because you really do not understand it.

What is ironic is that when this first appeared in some Sunday newspaper insert a number of math professors wrote in to respond to the columnist precisely as you did EP.  However they scolded her because they agreed with EP -incorrectly of course.  They suggested she purchase a basic textbook on probability and study it so that she would come to understand that switching would not help, since the probability was 'obviously' 1/2 that either remaining door had the big prize.  

The intuition that the odds are 50 to 50 that your original choice was correct (once a wrong choice is revealed) is very powerful.  It arises because of a bad analysis resulting from failing to take proper account of the significance of that wrong choice.  The probability when the wrong choice is revealed does not alter the probability you had at the outset.  I also was flummoxed when I first heard of the problem but also intrigued of course.

A similar problem which probably won't (but might) help is the three domino problem.  One has an X on both sides, one has an O on both sides and one has one X and one O on opposite sides.  You draw one domino from the bag and hold it facing you and see an X.  What is the probability that there is also an X on the other side?

The wrong analysis goes:  

There are two dominos I could be holding, the X-X or the X-O.
So the probability that there is an X on the other side is 1/2.

But the correct analysis is:

I could be looking at one of three X's.
There is another X on the back of two of those X's and an O on the back of the third one.
So the probability is 2/3 that there is an X on the other side.

It took me three reads to get this because you aren't describing dominos.  Dominos have  uniformly patterned backs like cards. Their faces have two different numbers or sets of dots on either side.  But you see the whole face. There are no xes.

But if you have slips of paper with x's or o's on either side it would work.