I really enjoy reading about those kinds of problems because they are so opposite to my intuition.
Even thinking about this more, and reading (and understanding) the solution, if I think about this, my guess would still be "well, with 182 people, half the pigeon holes would be taken, so there would be 50% chance the next person getting in the room would pick a taken hole".
Another similar problem is the Monty Hall problem. Simple, easy to understand when explained, but still, despite understanding the solution, doesn't feel right!
You're formulating the problem wrong in your mind.
Its true that if you have 182 people in the room all with unique birthdays and you add one more random person to the room there is a 50% chance of them sharing a birthday just like you described. But you're assuming you already managed to gather 182 people without any birthday collisions. That's a different problem then the one originally posed. In your case you collected 182 people with unique birthdays and checked the probability of a collision when adding one more. But the real question asks what is if you grabbed those 182 people at random what is the chance that any two of them already share a birthday?
Unfortunately I don't have any good intuition to share for the Monty Hall problem, as I haven't been able to get an intuitive understanding of it yet.
However I did just read about Bertrand's Box Paradox[1], and it's very much the same sort of thinking as the Monty Hall problem, but more intuitively understandable for me at least.
Even thinking about this more, and reading (and understanding) the solution, if I think about this, my guess would still be "well, with 182 people, half the pigeon holes would be taken, so there would be 50% chance the next person getting in the room would pick a taken hole".
Another similar problem is the Monty Hall problem. Simple, easy to understand when explained, but still, despite understanding the solution, doesn't feel right!