I may be wrong, but I feel the maths is wrong. You play the game 3 times, but the odd of repeating only happens 2 times: between the first to the second game, andbetween the second to the third game. So more 20%*20% → 4%. Furthermore, between the first and the second game, you are only counting one specific civilization repeating, but all could have been repeated.
I think we need to calculate (do not hesitate to correct me if I am wrong):
- The odd of having at least 1 repeating leader between game 1 and game 2.
- The odd of that leader came again in game 3.
For the 1, you have the maths
here. We need to test all civilizations coming back in the game, not only 1. By using 45 civilizations (instead of 48 in the link), the The answer is ~89.4%, not 20%.
For the 2, let's assuming the worst case scenario: only 1 leader is repeated. The odd of that leader coming back is 20%, like you said. I am afraid to not be good enough to calculate the real odd.
The odd of a civilization appearing three time in a row is at least 17.9% (~89.4% × 20%) in Large setting (9 AI).