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 one deck?

Numbers of each rummy

Set of 3 and set of 4: For any given rank, there are 4 sets of 3 and 1 set of 4 so this is just 4*13*12 = 624

Run of 3 and set of 4:  Every run will exclude a set in exacly 3 ranks so this is just 4*12*10 = 480

Set of 3 and run of 4:  Every run will partially exclude 4 ranks so this is 4*11*(9*4 + 4) = 1760

Run of 4 and run of 4:
Case 1 -- different suits is just 4*12*3*11 = 1584
Case 2 -- same suit the first run excludes possibilities for the second depending what rank it begins with.  This is
4*(7+7+6+5+5+5+5+5+5+6+7+7) = 280 The numbers in parentheses are the number of runs of 4 for each possible run of 3 beginning with ace low and ending with queen low.
Total = 1864

Grand total = 624 + 480 + 1760 + 1864 = 4728 possible rummies.

There are 52C7 = 133784560 rummy hands so the probability is

about .00003534 or about 1 in 28000.

One might expect this two happen about once every two years if four people played an average of 10 hands every night.

-Jer

Posted by Jer on 2005-01-07 18:36:07