Yes, you got it Henoch. I thought you were going to get it earlier because you were homing in on the episodes where odds were numerically expressed, and this one was one of a few that do that.
Hopefully the clues should make sense now. In summary, the odds were 10000 to 1, as quoted by Scotty and repeated by Kirk. These odds are between 7824.7 to 1 and "astronomical" which is usually 1 in a million or billion. Also, Riley mentioned that "one Irishmen is worth 10000 of you". I mentioned the word "matter" several time and thought that the word "opposite" might trigger the thought of antimatter, which needed to be precisely imploded to do a cold start of the engines. Spock raided the game because he knew of the known intermix formula that would give them the best chances. The risks were much less using this formula, with the main problem that it had never been experimentally tried.
Another interesting thing about this episode is that it is the only one that has the 3 regular ladies, Uhura, Rand and Chapel, all in the same episode. This clue was going to be mentioned if needed.
It's a decent episode if you are not finicky about your Irish tenors.