You want to compute 12.5/(12.5+56.25) instead.
Of course, I don't know how I missed that...
Now on to Monthy Hall the linked explanation is not that intuitive to me.
To me the intuitive explanation is that if I chose a goat, switching gives me 100% possibility to get a car and not switching 0%, if I chose a car, switching gives me 0% possibility and switching 100%, thus my original 2/3 chance to win a goat wins me a car with the same 2/3 chance if I switch.
I don't know if it is Bayesian what I am doing... let's play with 4 doors, 3 goats 1 cup, er, car. If I chose a goat, 75% chance, switching gives me an 50% chance so it is 37.5% if I chose a car, 25% chance, switching gives me 0%. So the switch gives me 37.5% in the four-door game. If I chose a goat, 75%, staying gives me 0%, if I chose a car, 25%, staying gives me 100%, that is just 25%. Still switch.
Am I doing it right? The line of reasoning being "If my prior is right... the evidence does this. If my prior is wrong, the evidence does that. Add it up." (probability of correct prior judgement probability of new judgement) + (probability of incorrect prior judgement probability of new judgement) something like this... and some of these four is apparently always 0 and 100% ?
Your argument does all the things that are necessary to solve Monty Hall, but it doesn't consider some things that could be necessary. (Now, maybe you would have realized that those things need to be considered, if they were necessary. I am just explaining how things can get tricky.)
Suppose instead of Monty Hall, we have Forgetful Monty Hall. Forgetful Monty Hall does not remember where the car is, so he opens a door (that you have not picked) at random, and luckily there is a goat behind it!
Here your line of reasoning still seems to apply: if you chose a ...
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