Confidential. Single Deck Deal up to 3 cards per hand. 132,600 possible combinations.
Odds of catching a natural 9 on the first card dealt is 13:1. Second card, Same. Third Card, Same. Chance of
catching 3 natural 9s is approx. 13 x 13 x 13. If catching a 9 is a winner, then, we must calculate how often a player will catch a natural or other 9. Define other 9: Any combination of 2 or 3 cards adding up to a toal of 9. Cards play at face value, ace is 1, and face cards zero or 10 - they don't count. Hence, player can win with 1, 2, or 3 cards. Player is dealt 2 cards at the outset of play. If either or both are a natural 9, he is a winner. Game over, unless he has two natural 9, then, he is given a 3rd card, to see if he catches a 3rd natural 9. Done. Note, if player's two cards are not natural 9s, they could add up to a total of 9. For instance, 8 and 1, 7 and 2, 6 and 3, 5 and 4, etc. if card is a face card or 10, to win, player needs, of course, a natural 9. If there is no 9, player gets 3rd card. It could be a natural 9. Or, it would mate with either of the two cards dealt (2 chances) or both cards (1 more chance to make a 9). Latter example: 2,3,4 = 9
This game wins 36.8% of the time, per the above rules. Try it with a deck of cards. The project: How often does the player win, and specifically, how often does the player win with 1 natural 9, two natural 9, or three natural 9. Also, how often, in this cycle, does player win with Three Cards, Suited, total 9 (like 1,2,6 all spades. Two Cards suited, total 9 (like 5, 4 hearts). 3 Cards Total 9. (like 1,1, 7) (note, presence of a face card does not count) Any 9. And, one special hand: 3+3+3 v=9. Finally, need to know how often player receives in the first two cards dealt Face Cards, in that two face cards = an auto losing hand. This is like 30.4% o the time. . . Once we have this info, we can project out for, say 1,000,000 hands, more.