Markov chain probability in a complete graph
The probability that you state is the probability that a random edge of the complete graph connects one of the $k$ nodes to one of the other $n-k$ nodes. This means that in your process, a random edge is chosen, and if exactly one vertex already knows the information, then the other vertex is made aware of it. Information spreads exactly when the chosen edge is of this form, which in your case happens with the probability you mention.