Right, thanks, the first 12 times. I actually only calculated 11, the correct probability is even less - 0.04%!
I think you have an extra zero in there. I calculate 0.414%, or once every 242 games.
I did notice another mistake. I had included the turn when they missed. So now I get 0.075%.
I calculated the chances of each of the 12 hits: 32%, 57%, 57%, 57%, 57%, 43%, 33%, 71%, 69%, 71%, 69%, 65%. And then I multiplied them. I don't know how our calculations differed.
I get 32%, 57%, 57%, 57%, 67%, 67%, 71%, 69%, 71%, 69%, 65%. So it seems you go wrong somewhere in the middle with 57%, 43% and 33% replacing the two 67%'s (from the 7/17's). You also only had 11 hits, not 12 -- the 12th line you listed was the first miss.
Ok, you're right, I put in the wrong numbers for turn 12 and 13. I don't know how I did that. But there were 12 hits, not 11, since my opponent hit twice on turn 13!
The correct percentages are: 32%, 57%, 57%, 57%,
67%, 67%, 67%, 71%, 69%, 71%, 69%, 65%. This makes the probability 0.28%, once in every 361 games.