So to get started , I make it clear that I do know how the Monty Hall problem works and although I had my good share of problems understanding it in the past , I did manage to come to terms with it on an intuitive level .
I know the explanations which go along the lines of "but two-thirds of the times you are more likely to pick the wrong door , so you should switch" or the one extending the problem to a 100 doors . I know the proof with Bayes theorem , also the ones with the tree diagram , and I do understand and acknowledge the reasoning behind all of them . But that's not what I am looking for.
The problem is :
Today I tried explaining my mother about this problem. I began with my usual "Say I pick door 1 , and the host reveals door 3 with goats behind it."
Then I say : "The reason you should switch is because when the host opens door 3 , the 1 in 3 probability corresponding to it gets fused with the 1 in 3 probability corresponding to door 2 , which makes door 2 have a two thirds chance of having the car" , which I believe is correct in all rational frames.
Now here's her rebuttal : The fusing that you explain , dear son , should happen equally between doors 1 and 2 and not just with door 2 , and that leaves me with a 50-50 choice.
Could someone explain to me in a direct manner , why the "fusing" of probability doesn't happen between both doors 1 and 2 but only with door 2 instead.
I tried to come up with one myself , but all of them are not very direct and convincing at the same time . I know its because the knowledge of the host influences the probabilities but I can't package it in a way that explains why the extra one-third chance gets accumulated on to door 2 . So I am asking for help here. I know its probably stupid given that I have other ways of doing the same thing , but that seems like I am dodging a crucial part that questions my understanding of this problem itself.