# Detour: The Overlapping Words Paradox

Explore a game called overlapping words paradox and see how it works.

## We'll cover the following

We illustrate the overlapping words paradox with a two-player game called “Best Bet
for Simpletons.” Player 1 selects a binary *k*-mer *A*, and Player 2, knowing what *A* is,
selects a different binary *k*-mer *B*. The two players then flip a coin multiple times, with
coin flips represented by strings of “1” (“heads”) and “0” (“tails”); the game ends when
*A* or *B* appears as a block of *k* consecutive coin flips.

STOP and Think:Do the two players always have the same chance of winning?

## Chances of winning

At first glance, you might guess that every *k*-mer has an equal chance of winning. Yet
suppose that Player 1 chooses “00” and Player 2 chooses “10.” After two flips, either
Player 1 wins (“00”), Player 2 wins (“10”), or the game continues (“01” or “11”). If
the game continues, then Player 1 should surrender, since Player 2 will win as soon as
“tails” (“0”) is next flipped. Player 2 is therefore three times more likely to win!

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