Counterintuitive examples in probability I want to teach a short course in probability and I am looking for some counter-intuitive examples in probability. I am mainly interested in the problems whose results seem to be obviously false while they are not.
I already found some things. For example these two videos:

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*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...

The Shooting Room Paradox

A single person enters a room and two dice are rolled. If the result
is double sixes, he is shot. Otherwise he leaves the room and nine new
players enter. Again the dice are rolled, and if the result is double
sixes, all nine are shot. If not, they leave and 90 new players enter,
and so on (the number of players increasing tenfold with each round).
The game continues until double sixes are rolled and a group is
executed, which is certain to happen eventually (the room is
infinitely large, and there's an infinite supply of players).
If you're selected to enter the room, how worried should you be? Not
particularly: Your chance of dying is only 1 in 36. Later your mother
learns that you entered the room. How worried should she be?
Extremely: About 90 percent of the people who played this game were
shot. What does your mother know that you don't? Or vice versa?

