The problem with the Monty Hall is that there is a hidden rule that people never explain. If you put there rule there explicitly, it's reasonably easy to see that the presenter is messing with your up-front probabilities when he opens a door.
When the problem was originally going around, this wasn't so hidden. There was a cultural awareness of who Monthy Hall was and experience with the show Let's Make a Deal, and that he never showed the prize door (although, he was not bound to the presented rules). Although, personally, I remember being explicitly told that the host knew where the prize was and so knowingly opened a non-chosen, non-prize door.
My brain still wants to short circuit to two doors, one prize, random selection, 50% chance.
> My brain still wants to short circuit to two doors, one prize, random selection, 50% chance.
This seems to be a wide-spread belief/bias ... if there are two choices, two options, two outcomes, then it's 50:50.
OK, I buy a lottery ticket ... either I win or I lose, so it's 50:50, right?
Of course not. If you can start to see, by default, that two options are not usually equally likely, then you start to lose this misconception.
But really, you always have this situation: the prize is behind the door you chose, or it's behind one of the other two. That always remains true, so your probability of having the prize is always 1/3, even if you get to see behind one of the other doors. You're still sticking with your original choice, or switching to both the other doors (although one of them is open, so you ignore it).
Exactly: what counterintuitively changes the overall probability of picking the correct door is Monty’s introduction of new information that biases the contestant to the right choice. Many years ago I was in such disbelief about this result that I wrote a little Monte Carlo simulation to “prove” I was right. In fact, it proved I was wrong!
The original problem formulation lacked it. Also this comic lacked it. Most people who describe the problem doesn't include the titbit. It is a very misunderstood point, you can even see it everywhere in this thread.
You're entirely right, and in your other comment as well. I just recognized this was about Monty Hall and didn't even bother reading the details. I also assumed other people would do the same: mentally invoke the entire paradox. It is indeed true that someone discovering the paradox with this comic wouldn't have nearly enough info to understand it, and would be correct to say it's wrong.
Based on him knowing where the prize is, it’s hard to escape the idea that he’s only showing you a new door because you picked the right one, therefore intuitively you feel like you should stick with your guns. At least if you’re wrong the way you chose, you won’t feel like you’ve been manipulated.
