The role of probability in solving the Monty Hall problem

[Jenny A]

There are no patterns to consider. One door always has the prize, and the others don't.  Choose one door and you have a one third chance of winning.  Choose two doors  and you have a two thirds chance of winning.  After Monty's reveal switching gives you effectively two rather than one doors.  Staying with the first door leaves you with  a one third chance.


EDIT:  Look around it this way. Suppose you had the choice of choosing one door or choosing two doors initially  with the understanding that Monty would show you a goat behind one of the two doors you choose and you'd get the remaining door of your two door choice.  That's what's actually happening if you switch.

Then the choice has been 50/50 all along, and it doesn't change in the least. You're just either in on it or not, that's all that changes. The chances of you being right, however, are the same throughout.

At the start of the game, running the original experiment, from your perspective you have a 1/3 chance of getting right. From the moderators' perspective and the one that is actually right since he has all the information you have a 50/50 chance. So this is not so much a puzzle as it is a trick. No mystery whatsoever here, the chances stay the same from the beginning, only your understanding of them improves, not so the actual chances.

Wrong.  Monty is required to show you a goat you didn't choose. Those are the rules. 

At the start of the game one of two things will happen.  One,  you will choose the winning door.  There is a one third chance of that happening.  In that case Monty will show you one of the goats because that is all he can show you. If you switch you will get the other goat and lose.  But chances are two out of three that you will choose a goat to begin with and  then Monty must show you the other goat because those are the rules.  In that case, if you switch, the other door is the prize. So, if you switch you win two out of three times.

If you don't switch,  your chances remain one in three because that is your chance of picking correctly in the first place.