I'm pretty sure somewhere your math is going wrong, but that's not really relevant because I don't think the hard probabilities are particularly relevant to my point anyway. We've established that if the surgeon does 10000 surgeries with 50% success but then does 20 with 100% success then that's not luck, that's skill. Just for a second I'm going to bend the original statement.

Instead of the last 20 being 100% the surgeon some point has done 20 successful surgeries in a row. Let's say he did 5000 surgeries with 50% success, then did 20 with 100% success and then did the next 5000 with 50% success. Would you still call that skill or is it now luck? I think it would be misleading to call it skill because their success rate didn't change after the 20 surgery streak.

But when we put those 20 to the end it becomes skill? So just because we don't know the success rate of future surgeries we're supposed to believe he's better than 50%? Call me a skeptic but it doesn't really matter how probable or improbable those 20 surgeries are, I wouldn't consider that an indication of skill. If someone flipped tails 20 times in a row I wouldn't go "wow, what a skilled coin flipper"