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 seven-card rummy with two decks?
- Playing ten-card rummy with one deck?
- Playing ten-card rummy with two decks?
- One of the cards was inadvertantly dropped on the floor before dealing for seven-card rummy?

- Playing seven-card rummy with two decks?

Set of 3 and set of 4  The sets may be in the same suit so there are two ways of getting the second set. 
56*13*(70*12 + 5) = 615160

Set of 4 and run of 3  Beginning with the run, the set may be in either the different or same suit.
12*8*4*(3*10*70 + 1*3*35) = 846720

Set of 3 and run of 4  Beginning with the run, the set may be in either the different or same suit.
11*16*4*(3*9*56 + 1*4*35) = 1163008

Run of 3 and run of 4
Case of differing suit:
12*8*4*11*16*3 = 202752
Case of same suit:  The runs may overlap to varying degrees depending of where the run of 3 begins for example if it begins with the A there are 2+4+8+16+16+16+16+16+16+16+8 = 134 possibilities for the run of 4.  Altogether, then there are
8*4*(134+128+116+108+108+108+108+108+108+116+128+134) = 3328

Grand total = 2830968

There are 104C7 = 21243342120 possible hands so the probability is about .000000108462

I'm wondering why this is so much smaller.  Maybe it is not right.

-Jer 

Posted by Jer on 2005-01-07 19:16:50