First off, I don't think its 50/50 always. That's not what this is about.
I know that simulations show that if Monty randomly picks a goat, it IS 50/50. I always thought I understood why this was, but recently I've been looking at discussion boards and the explanation doesn't seem to match mine.
I always thought that if Monty chose a goat randomly, the probability was actually more complicated. It ends up being 50/50, but there are more possible scenarios.
However, I've been seeing lots of explanations that go something like this: "If Monty purposefully reveals a goat that means that he's given you more information, but if he's chosen randomly he tells you nothing." It has me wondering if maybe I've overthought it. Is there actually a more simple explanation for why a randomly revealed goat gives you 50/50?
**By "randomly chooses a goat" I mean that either Monty has no idea what is behind each door or that if he does know he ignores this knowledge in favor of a random choice (ie. coin flip determines which door he opens). Monty only has a choice between the two doors that you have not chosen. Sorry I should have been clearer.
**** Okay let me put it this way. Everyone says that if you choose at random you get no new info. What I don't understand is why this is true. Why does the random reveal not also give you more info? It seems to me that it does..