Suppose you play the following game: There's a certain buy-in, and at every turn you flip a coin. If anytime you flip a tail, you lose the game and leave with your winnings. If you flip a head on the first flip, you win $\$1$. If you flip heads on the second flip, you get $\$2$, on the third flip $\$4$, and so on. Now, if a casino were to host this games, how much should they make their buy-in?
Intuition says not much, but mathematically they should make it as high as they want. Why? Because the payout is infinite. The probability of flipping heads on the first flip is $\frac 12$, which gives a $\$0.50$ average payout. The probability that you flip heads on the second flip (which also means heads on the first flip) is $\frac 12\times\frac 12=\frac 14$, which also pays out $\$0.50$ on average. Continuing like this gives you a payout of $\sum^{\infty}\$0.50=\$\infty$ every time you play the game! Not such a bad thing, but it leads to my main question (shortly after).
Suppose you hold a party with $30$ people in it, and you want to find the probability that any two of them will have a birthday on the same day. Do you expect that to happen, or not?
Again, common everyday intuition says it seems unlikely that any two people out of thirty will have a birthday on the same day, but, again, mathematically, it is more likely than not. The exact probability is $1-\frac{365!}{365^n(365-n)!}\approx 0.7063$. So is it time to ask the question?
Why do some mathematical ideas seem counter-intuitive? Mathematics isn't based off of physical observations; it's an abstract concept, so shouldn't it explain our world better, not worse?
The above game (which I was told is St. Petersburg paradox) is only an example of what I mean when I say "counter-intuitive". Among others, ones I can name off the top of my head are the Monty Hall problem, Benford's Law, and the Banach-Tarski paradox. Those all have specific aspects to which a normal non-mathematician would turn their heads in confusion.
I really hope my question isn't too philosophical for this site.
This question has been in my head for as long as I can remember, so I decided to post some of my thoughts. Mathematical laws don't just hold for our world or our universe. It holds for all universes. For example, maybe the Banach-Tarski paradox makes perfect sense in $34$ dimensions. Or maybe the second dimension finds the concept of $\pi$ being irrational hard to grasp, whilst we find it easy. The most important thing to note is that mathematics is always right. It doesn't matter what we think. We're stupid. But in the long run, math has and always will get out on top.
Is the reasoning in the previous paragraph correct? The answers so far are good, but they don't really address counter-intuitivity in general, instead specific problems. Several answers below state something along the lines of "some ideas seem counter-intuitive because we've adapted to it; that is to say, it is best for the human race". Can any of you think of a practical application of counter-intuitive ideas in the evolution of humankind? I certainly can't.
So what do you think? I know my question doesn't have a solid answer, and I know it might be put on hold because of it (please don't though!). I just want to put my question out there, and hope it gets answered.
Thanks for reading!