## Amusing Cribbage Facts

Have you ever wondered about the distribution of Cribbage scores for a hand pulled at random (with four cards in the hand and one card up)? Neither have I. But hey, this is the Web, and you never know when someone might want to know something like this.

Picking 5 cards from a 52-card deck gives roughly 2.5 million possible hands. Since exactly one card will be the up-turn, we need look at a mere 12,994,800 hands. My Sparc scoffs at such tiny numbers.

### Rules Used for Scoring

15's
Each combination of cards summing to 15 (where face cards are worth 10) counts 2 points.

Pairs (3-of-a-kind, etc.)
Each pair counts 2 points (hence 3-of-a-kind counts 6 points).

Runs
Each run of 3 or more cards (e.g., 5-6-7) counts the length of the run in points. Duplicate cards in the run count once for each combination, so A-A-2-3-3 counts 12 points as a quadruple run of 3 (not counting scoring from pairs). Aces are low.

Flushes
4-card flushes (in the hand only) count as 4 points; 5-card flushes count as 5 points.

His Nobs
A Jack in the hand of the same suit as the upcard counts 1 point.

### The Distribution

The various columns represent the score of the hand, the number of hands with that score (out of 12,994,800), and the percentage of such hands.

 0 1009008 7.7647 10 388740 2.9915 20 8068 0.0621 1 99792 0.7679 11 51680 0.3977 21 2496 0.0192 2 2813796 21.6532 12 317340 2.4421 22 444 0.0034 3 505008 3.8862 13 19656 0.1513 23 356 0.0027 4 2855676 21.9755 14 90100 0.6934 24 3680 0.0283 5 697508 5.3676 15 9168 0.0706 25 0 ------ 6 1800268 13.8538 16 58248 0.4482 26 0 ------ 7 751324 5.7817 17 11196 0.0862 27 0 ------ 8 1137236 8.7515 18 2708 0.0208 28 76 0.0006 9 361224 2.7798 19 0 ------ 29 4 0.00003

### The Code

Do you really think I'd make such difficult code available for public consumption? Am I prepared to have people point out bugs and poor coding tendencies? Do I want to hear about how many more comments I need to achieve an optimal comment-to-code ratio? Do you enjoy reading lots of silly questions?

### Acknowledgements

My officemate, Seth Copen Goldstein, wanted to receive credit as a co-author for this work (don't ask me why!), so...there it is. And despite his better judgment, Mike Mitzenmacher filled me in on the details of Cribbage rules (which I had partially forgotten).

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