Cribbage is a game in which each of two players is dealt a hand of six cards. Each player then discards two of his six cards to a four-card "crib" which alternates between players. After the discard, the top card in the remaining deck is turned up. The top card is then ignored until the counting round, unless it is a jack, in which case the dealer pegs two points for "heels." Cards are then alternately played out by the two players, with points being scored for pairs, 3s and 4s of a kind, runs, cumulative total of 15 and 31, and playing the last possible card ("go") not giving a total over 31. All face cards are counted as 10 for the purpose of playing out, but the normal values of , , are used to determine runs. Aces are always low (). After all cards have been played, each player counts the four cards in his hand taken in conjunction with the single top card. Points are awarded for pairs, 3s and 4s of a kind, runs, and combinations of cards giving 15. A jack having the same suit as a top card is awarded an additional point known as "nobbs." The crib is then also counted and scored. The winner is the first person to peg a certain score, as recorded on a cribbage board.

The best possible score corresponds to three 5s and a jack in the players hand with a top 5 the same suit as the jack, giving four 5s and a jack available for counting. This hand has pairs of fives (score 12), triplets of five totaling 15 each (score 8), four 5s available for pairing with the jack totaling 15 (score 8), and 1 point for nobbs, giving a total score of for the hand.

All scores from 0 to 29 are possible, with the exception of 19, 25, 26, and 27. For this reason, a hand scoring zero points is sometimes humorously referred to as a "19-point" hand. The following table summarizes the possible scores (Sloane's A143133), their frequency (out of five-card hands, counting the top card as the fifth), and their probabilities. The average score from a random hand is 511661/108290, or about 4.7249. Of course, these probabilities assume that the cards discarded to the crib are effectively chosen at random. Since they are presumably not, better scores are actually more likely than indicated in this table.

score	frequency	probability
0	1025024	0.07888
1	101376	0.007801
2	2843456	0.2188
3	510528	0.03929
4	2874672	0.2212
5	704208	0.05419
6	1787376	0.1375
7	754144	0.05803
8	1118048	0.08604
9	353352	0.02719
10	375208	0.02887
11	42112	0.003241
12	308480	0.02374
13	15232	0.001172
14	87436	0.006729
15	8576	0.0006600
16	57272	0.004407
17	11192	0.0008613
18	2248	0.0001730
20	7828	0.0006024
21	2472	0.0001902
22	444	0.00003417
23	356	0.00002740
24	3680	0.0002832
28	76
29	4

Sloane, N. J. A. Sequence A143133 in "The On-Line Encyclopedia of Integer Sequences."

CITE THIS AS:

Weisstein, Eric W. "Cribbage." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cribbage.html