The Mathematics and Odds of Gin Rummy

Posted in Mathematics of Gin Rummy

When gin rummy became really popular in the 1970’s, there were some players who decided to do a little computer work on the actual mathematics and odds of gin rummy. They are extremely interesting, albeit rather unimportant. However it does give you an idea of the possible hands and odds during the game.

For instance, there are 15,820,024,220 possible ten-card hands in gin rummy. The non-dealer, who receives an eleven-card hand, has the possibilities increased to 60,403,728,840 different hands. After the deal, the combinations and permutations for either hand must be based on 41 cards. Actually each player can start with one hand out of a separate total of 1,471,442,973. After the cards have been dealt, the number of possible hand-against-hand combinations climbs to a whopping 2,165,144,422,791,078,729! Therefore, to give the probabilities and odds of each of the various possibilities in gin rummy would be an exercise in futility and of little value to the player in the game. There would be just too many figures to remember, but there are certain valid general principles that all expert players keep in mind.

We already know the mathematics of gin is based primarily on the law of probabilities. For instance, the probabilities of the dealer being dealt a three-card meld in his first 10 cards are about two out of every five hands, while the chances of the non-dealer being dealt a run in the first 11 cards that he sees, are approximately one out of two. The odds of the non-dealer throwing a card that his opponent can use in a meld in his first play are:

If the discard is a: Odds: Probability:
King or Ace 6 to 1 against 0.141
Queen or Deuce 5 to 1 against 0.171
Any other card 4 ½ to 1 against 0.182
. . .

These figures are based on the opponent’s using the card immediately in a meld, and not on his considering the card as an improvement to his hand, or as reducing his total of unmatched cards.

As play in a hand progresses, the chances of picking a wild discard increases greatly. For instance, in the late stages of a hand, it is almost certain that a wild card will be picked up. However, at the beginning of a hand, you do not know the cards that are in the stock and probabilities depend on the number of ways in which the discard can be used. These figures are as follows:

Usable Ways: Odds: Probability:
3 sequences and 3 sets 3 ½ to 1 against 0.212
2 sequences and 3 sets 4 to 1 against 0.194
1 sequences and 3 sets 5 to 1 against 0.171
3 sequences and 0 sets 7 to 1 against 0.124
2 sequences and 1 set 7 to 1 against 0.129
2 sequences and 0 sets 11 to 1 against 0.084
1 sequence and 1 set 10 to 1 against 0.091
1 sequence and 0 sets 20 to 1 against 0.047
0 sequences and 1 set 20 to 1 against 0.047
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