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.
This is really cool. I'd love to dive right into this and get some statistics going. I'm actually working on adv. statistics with my professor on a private research venture, and I'd love to use this experience as a top Plus