Back 2 Back keno

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I'm wondering if somebody could help me with a math question for the video keno game: Back 2 Back Keno. In this keno game, you get paid for keno wins in the usual way according to how many spots you mark, how many you hit and how much you bet (the pay tables vary a little bit for max betting vs. bet 1 and 1 < bet < max). In addition, if you win a game, you have a chance to win a "back 2 back" pay in the next game if you hit enough of the numbers you hit in the game before.

Example: in the first game, you mark 6 spots, and hit 3 and it pays 2 times your bet. On the next game, you play the same numbers, and if you hit the same 3 numbers (and no other winning numbers), then you'd get two pays: 2 times your bet (for hitting 3 this game), and 27 times your bet (for hitting the same 3 numbers this game that you hit in the game before).

So my question is: given you marked 6 spots and hit 3 this game, what are the odds of hitting those same 3 numbers the next game? More generally, how do I compute those odds for all the various mark X, catch Y, and catch Y back 2 back scenarios.

Thank you for your assistance!

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