TITLE: Counterintuitive examples in probability
QUESTION [192 upvotes]: I want to teach a short course in probability and I am looking for some counter-intuitive examples in Probability. The results seem to be obviously false but they true or vice versa.
I already found some things. For example these two videos:

Penney's game
How to win a guessing game

In addition, I have found some weird examples of random walks. For example this amazing theorem:

For a simple random walk, the mean number of visits to point $b$ before returning to the origin is equal to $1$ for every $b \neq 0$.

I have also found some advanced examples such as Do longer games favor the stronger player?
Could you please do me a favor and share some other examples of such problems? It's very exciting to read yours...

REPLY [2 votes]: This was grazed both in the original post and in the comments, but I'd like to flesh it out a bit.
Polya's Theorem
The simple random walk on $\mathbb Z^d$ is recurrent iff $d < 3$. Restated:

A random walk on $\mathbb Z$ started at the origin will eventually return to the origin (with probability $1$).
A random walk on $\mathbb Z^2$ started at the origin will eventually return to the origin (with probability $1$).
A random walk on $\mathbb Z^3$ started at the origin may return to the origin, but it also may not; the probability of an eventual return is about $0.35$.

Expected hitting time
Although a random walk on $\mathbb Z$ or $\mathbb Z^2$ that starts at the origin is guaranteed to return to the origin as above, the expected time to do so is infinite.
NB: This is a fairly elementary result in Markov Chain theory, so whether it is surprising is obviously up to the beholder. This is my favorite example of a random variable that is finite with probability 1, yet has infinite expected value.
Penney's Game, revisited
The Numberphile video explored the nontransitive nature of Penney's game; that is, for any 3-coin sequence I choose, you could respond by choosing a 3-coin sequence that is likely to appear first. There is another feature of Penney's Game that I find similarly counterintuitive: The expected wait time is not the same for all sequences of 3 coins. This is not a statement about the competitive likelihood of finding one before another, but rather a statement about the expected number of throws required to encounter a sequence in isolation. Specifically:

HHH / TTT require 14 flips on average
HTH / THT require 10 flips on average
The other 4 combinations require 8 flips on average