We can't transmit information instantaneously, but there are nevertheless certain distributed tasks we can do better at when we have a source of entangled states.
To paraphrase, Alice, Bob and Carol play a game where they can't communicate (after deciding on a strategy) and the referee shows each of them a bit. After being shown the bit they must reply with 0 or 1. The four possible combinations the referee chooses uniformly from is:
(Alice: 0, Bob: 0, Carol: 0)
(Alice: 1, Bob: 0, Carol: 1)
(Alice: 1, Bob: 1, Carol: 0)
(Alice: 0, Bob: 1, Carol: 1)
First, without using probability, if A0 is the answer Alice gives when shown 0 and A1 when shown 1 (and similarly for Bob and Carol), you get the following set of equations needed to win always:
A0 + B0 + C0 = 0 (mod 2)
A1 + B0 + C1 = 1 (mod 2)
A1 + B1 + C0 = 1 (mod 2)
A0 + B1 + C1 = 1 (mod 2)
Now, probability doesn't help here, because any mixed strategy can be shown to be equivalent to a combination of pure strategies, none of which can guarantee a win.
Yet with a shared entangled state, it can be won 100% of the time. This isn't a probability thing - you can actually guarantee a win using the quantum strategy, even over arbitrarily large distances.
As an even simpler example, let's say Alice and Bob need to solve a really hard problem that can be cleanly split into two parts, A and B. In round 1 Alice and Bob can't communicate but they have access to an entangled photon. In round 2 they meet and compare notes. If they choose which part to work on randomly, then in only 50% of the outcomes are both parts A and B solved when round 2 starts. If, on the other hand, they use the polarity of the entangled photon to decide which part to solve, then quantum mechanics guarantees that they both solve different parts. It's random and unpredictable which part gets assigned to whom, but in 100% of outcomes when round 2 starts, Alice has solved one of A or B, and Bob has solved the other one.