It took me a long time to figure out the Monty Hall problem. I was relieved to discover it also took Erdos a long time, and most others. After much stumpification, I found this thread and finally got it:
The exact wording of the problem can change the answer. For example, in this version
There are two doors with goats and one with a car. You choose one door from the three. The host selects one of the doors with a goat from the remaining two doors, and opens it. Should you switch doors if given the chance?
has a different answer than this one
There are two doors with goats and one with a car. You choose one door from the three. The host chooses one of the remaining two doors at random and opens it, showing a goat. Should you switch doors if given the chance?
In the first example the host's choice is either forced (if you chose a goat initially) or doesn't make a difference (if you chose a car).
In the second example, the host could have opened a door with a car because he was choosing the doors at random. Given that he didn't, your estimate of the probability that you have chosen the car should increase - in fact, it should increase to 1/2, and switching doesn't make any difference. (https://news.ycombinator.com/item?id=8663349)
So I contemplated why I had a hard time, why most people do. I contemplated why pigeons figure it out more quickly than we do, and I think it has something to do with ego. I know that the ego does everything it can to eliminate its own self-awareness (even as it paradoxically attempts to redirect all awareness to itself), and I'm guessing that this is a case where our ego hides our role: we overlook the fact that our choosing the door has affected the host's choice, and this causes us to assume his choice was random..?
Just a thought...