A simple setting for demonstrating the usefulness of entanglement involves a two-player game known as the CHSH game. The game is a variant of an experimental setup (by Clauser, Horne, Shimony and Holt) that is often used to illustrate Bell's theorem.
We shall call the two players 2 players Alice and Bob. We will also have Charlie as a referee that decides if Alice and Bob wins the game. They can decide on any strategy before the game commences but they cannot communicate with each other once the game starts.
To begin, Charlie picks two uniformly random bits $x$ and $y$, and gives $x$ to Alice and $y$ to Bob. Alice answers the referee with bit $a$, while Bob replies with bit $b$. After getting $a$ and $b$, Charlie checks whether
$a \oplus b = xy \mod{2}$,
that is, that the XOR of the output bits $a$ and $b$ is equal to the AND of input bits $x$ and $y$. If so, then Alice and Bob win the game.
