TITLE: Examples of simple but highly unintuitive results?
QUESTION [24 upvotes]: QUESTION: What are some simple math problems whose answers are highly unintuitive, and what makes them so?
There are plenty of unintuitive and frankly baffling results in math, like the Banach-Tarski paradox and this crazy MSE question which uses the axiom of choice to predict real numbers. However, these are pretty esoteric, and a layman might have trouble even understanding what exactly the question is asking. I’m more interested in examples like the Potato paradox:

Fred brings home $100$ kg of potatoes, which (being purely mathematical potatoes) consist of $99\%$ water. He then leaves them outside overnight so that they consist of $98\%$ water. What is their new weight? The surprising answer is $50$ kg.

I think I can explain why this answer seems unintuitive. Our intuition tells us that a small change in the water percentage should result in a small change in the mass of the potatoes. However, this heuristic is misleading in this case, in part because of the fact that $1/x\to \infty$ as $x\to 0$ and $1/x$ makes large “jumps” in value for $x$ close to $0$.
What are some other examples of simple problems with unintuitive answers? (I expect that there are plenty of examples that have to do with probability, since humans have terrible probabilistic intuition, and plenty of examples involving infinity, since people have a hard time conceptualizing the infinite.)
Also, please try to articulate exactly why you think your problem has an unintuitive answer, as I’ve attempted to do for the Potato paradox.

REPLY [2 votes]: In a far future, humanity has settled on Mars and Venus. Now Earth gets visits both from Martians and from Venusians, and some of them happen to be criminals.
In particular, of all the crimes done by visitors from other planets, $3/4$ are done by Martians, while only $1/4$ are done by Venusians.
Therefore a Martian visitor is three times as likely to be a criminal than a Venusian visitor, right?
Wrong! With the data given above it is impossible to say whether Martian or Venusian visitors are more likely to be criminals.
The conclusion would only be right if there happen to be as many Martian visitors as Venusian visitors, and moreover Martian and Venusian criminals do on average the same number of crimes on visit.
If there happen to be three times as many Martian visitors as Venusian visitors, both have the same crime rates. Should the number of Martian visitors be five times as high as the number of Venusian visitor, the same numbers would even mean that Martian visitors are doing less crimes on average than Venusian visitors. On the other hand, if Venusian visitors outnumber Martian visitors, it means that Martian visitors are even more criminal than those numbers suggest.
Also, if each Martian criminal on average tends to do three times as many crimes as a Venusian criminal, then even with equal numbers of visitors it means that the probability of a random Martian visitor being a criminal is the same as the probability  of a random Venusian visitor being a criminal. In that case the Martian criminals do more crimes not because there are more of them, but because the criminal ones are more active.
This is unintuitive because most people don't even notice that they make hidden assumptions when making that wrong conclusion.