# Communication complexity of reconstructing a random bit-string of length $n$

This seems like a folklore claim but I cannot find any reference to it. If Alice has a bit-string of length $$n$$ where each entry is independently set to 0 or 1 equiprobably, and Bob's goal is to reconstruct the string with a success probability at least 0.9. What is the simplest way to show that the randomized communication complexity (multiple rounds are allowed) is $$\Omega(n)$$?

One can use an overkilled proof: such a protocol will lead to a protocol for Disjointness and we know that Disjointness requires $$\Omega(n)$$ bits of communication.

Suppose Alice always sends exactly $$k$$ bits to Bob during the protocol. On average, how many possible candidates for her $$n$$-bit string are consistent with the communication transcript? What does that tell you about the probability that Bob guesses which one of those candidates is the correct one?