Monty Hall Problem Revisited

I mentioned in a previous post that I don't fully "get" the solution to the Monty Hall problem.

An friend of mine, John, didn't accept that I couldn't understand this and sent me an email about it.

On thinking about it again I think I do get it now.

My problem was this: intuitively I think that once Monty shows you one of the wrong doors, you imagine that you are faced with two doors (a "stick or switch" choice) so your changes are 50:50. So why should you switch?!

But intuition is doing you a disservice here.

Your changes of getting the right door from the original 3 were 1 in 3. You will only have the right door one time in 3 on average. Your changes if you stick are not 50:50 - they are still 1 in 3!

The chances that one of the other doors is the right one are therefore 2 in 3. But since Monty has shown you a wrong door, your changes if you switch to the remaining door are therefore 2 in 3.

So my intuition that Monty showing you a wrong door has changed the odds of you being right (by either sticking or switching) to 50:50 is the mistake I was making.

If Monty gave you the opportunity to change your selection without opening another door, there would be no reason to do so: the odds would still be 1 in 3.

But since he knows the right door, and is therefore able to open a wrong door, he has changed the dynamic. He has compressed the 2 in 3 odds into a single door!

So your chances of moving OFF the right door (which you had all along) are 1 in 3.

But your changes of moving TO the right door are 2 in 3.

So you should move!

Thanks John: I think I get it now!

Monty Hall Problem

I like to think of myself as a fairly smart guy. I like to think that I have a very logical mind. I like to think that I can work most things out, given enough time.

But there is one thing that I cannot get my head around: the Monty Hall problem.

The idea is simple: a game show host (Monty) as you to pick one of three door to get a big prize. You pick one of the doors (say A). He then opens another door (say B) which does not reveal the car and asks if you would like to change your choice to the other door (C) or stick with the one you have chosen.

When presented with this situation, most people will not change. Why should they?

But the best option here is to change. Doing so changes you chances of winning from 1 in 3 (picking one door from three) to 1 in 2 (picking one door from two).

My mind understands this. On some level.

But there is another part of my mind which cannot see how changing from one of the three doors to another one of the three doors could actually impact the likelihood of winning!

I have been aware of this conundrum for some years so I have concluded that my mind will never be able to full wrap itself around this one.