Imagine a single round round-robin chess tournament of 16 players. What is the probability that any two players meet each other within first 3 rounds?

Question

I am thinking of doing something like this:

P(Player A and Player B meet in the first round) + P(Player A and Player B don't meet in the first round)*P(Player A and Player B meet in the second round) + P(Player A and Player B don't meet in first round)*P(Player A and Player B don't meet in the second round)*P(Player A and Player B meet in the third round)?

Am I thinking it correctly? Would like to hear what other think?