Are chess games repeated

Question

If chess games, in the history by all players, were played randomly, could we have repeated games?

Shannon number is a lower bound for all possible chess games which is $10^{120}$. Suppose that we already had $10^{15}$ chess games (I have no idea if this number is accurate) in the history, could they repeat?

Thanks

Edit: Let me add that we eliminate short games. In fact my question is: If you choose a number from a set of $10^{120}$ and repeat that for $10^{15}$ could you obtain repeated numbers?

Answer 1

Hint: What is the probability that a game is given by the two move checkmate 1.f3 e5 2. g4 Qh4 #

Answer 2

Yes, under these assumptions, there are very likely some repeated games. The reason is that there are some very short chess games. For instance, Fool's Mate ends in 4 half-moves. Each of those 4 half-moves has under 100 options, so the probability of any one game following that path is at least $10^{-8}$. With $10^{15}$ games, you are very likely to have had at least two such games.