In the card game of Rummy, all players start with the same number of cards and the aim is to fill your hand such that all cards are in exactly one meld. Each individual meld is composed of 3 or 4 cards and can each can be completed two ways: cards of the same number/court or consecutive cards of the same suit. (This would mean you have a meld of 3 and a meld of 4 in 7 card rummy and 2 melds of 3 and a meld of 4 in 10 card rummy.) Each individual ace can count as higher than a king or lower than a 2, but not both. (This means K, A, 2 is not allowed.)
What are the probabilities of being dealt a winning hand when: (Note that all decks are without jokers)
 Playing seven card rummy with one deck?
 Playing sevencard rummy with two decks?
 Playing tencard rummy with one deck?
 Playing tencard rummy with two decks?
 One of the cards was inadvertantly dropped on the floor before dealing for sevencard rummy?
I'll use some poker terminology, but this is what I see for the first casesevencard rummy with one deck:
Two "straight flushes" same suit: (2*(C(9,2)1))*4 = 280 combinations of cards.
(The *4 is for the 4 suits, the C(9,2) is for laying out A,2,3,...,Q,K,A, and taking a block of 3 as a straight and a block of 4 as a straight, so calling individual cards I, the 3straight as 3 and the 4straight as 4, one example is III3II4II, and we're choosing 2 out of the 9 to be the 3 and the 4. It can't be 3IIIIIII4 as both straights cannot include the A, thus the subtraction of 1. That whole thing is doubled to account for the possibility of the 4straight being lower down than the 3straight.)
Two "straight flushes" of different suits: 11*10*4*3 combinations of cards. (The 11 is the number of face values the 3straight can begin on; the 10 is the number of face values the 4straight can begin on; the 4*3 is the number of possible pairs of suits.
Threeofakind and 4ofakind: 13*4*12 combinations of cards. (13 choices for the 3ofakind denomination times four possible missing suits times 12 choices of denomination for the 4ofakind for each of the choices of the 3ofakind.)
3straightflush and 4ofakind: 11*4*10 combinations (11 choices for denomination beginning the 3straight times four possible suits times ten choices of denomination for the 4ofakind  as three denominations are unavailable in a given suit).
4straightflush and 3ofakind exclusive suit: 10*4*13 (10 choices for denomination beginning the 4straight times four possible suits times 13 choices for 3ofakind).
4straightflush and 3ofakind inclusive suit: 10*4*9*3 (10 choices for denomination beginning the 4straight times four possible suits times 9 choices for 3ofakind denomination time 3 choices for 3ofakind suit).
This totals to 280+11*10*4*3+13*4*12+11*4*10+10*4*13+10*4*9*3 = 4,264 combinations, out of the C(52,7) = 133,784,560 possible hands or a probability of about 1 in 31375 or about 0.000031872.
As to the last part, it shouldn't make a difference if a card had been dropped on the floor before dealing. It's like any other random way of that card not being in the dealt hand.

Posted by Charlie
on 20050107 18:35:43 