TITLE: Common false/unproven "popular mathematics" claims
QUESTION [7 upvotes]: Motivated by the common unproven claim in memes that every finite sequence of digits appears in the digits of $\pi$ (see Does $\pi$ contain all possible number combinations?), I started wondering what other similar common claims are out there?
Criteria

A claim about math that's "commonly" stated as true by math enthusiasts who are non-mathematicians. For instance, something that one might find posted somewhere as a meme.
The claim must be either false or unproven.

REPLY [3 votes]: Let's assume playing with a fair coin.

Common claim (false)
In a long coin-tossing game each player will be on the winning side for about half the time, and the lead will pass not infrequently from one player to the other.

Contrary to this false claim we have with probability $\frac{1}{2}$ that no equalization occurs in the second half of the game regardless of the length of the game. Furthermore, the probabilities near the end point are greatest.
This is a consequence of the Arc sine law for last visits.