Cribbage statistics

In cribbage, the probability and maximum and minimum score of each type of hand can be computed.

Main article: Cribbage

Distinct hands

• There are 12,994,800 possible hands in Cribbage: 52 choose 4 for the hand, and any one of the 48 left as the starter card.

${\displaystyle {52 \choose 4}\times 48=12,994,800}$

• Another, and perhaps more intuitive way of looking at it, is to say that there are 52 choose 5 different 5-card hands, and any one of those 5 could be the turn-up, or starter card.
  Therefore, the calculation becomes:

${\displaystyle {52 \choose 5}\times 5=12,994,800}$

• 1,009,008 (approximately 7.8%) of these score zero points,[1] or 1,022,208 if the hand is the crib, as the starter must be the same suit as the crib's four cards for a flush.
• Not accounting for suit, there are 14715 unique hands.[2]

Maximum scores

• The highest score for one hand is 29: 555J in hand with the starter 5 of the same suit as the Jack (8 points for four J-5 combinations, 8 points for four 5-5-5 combinations, 12 points for pairs of 5s and one for his nob).
• The second highest score is 28 (hand and starter together comprise any ten-point card plus all four 5s, apart from the 29-point hand above).
• The third highest score is 24 (A7777, 33339, 36666, 44447, 44556, 44566, 45566, 67788 or 77889).
• The highest score as a dealer from the hand and crib is 53. The starter must be a 5, the hand must be J555, with the Jack suit matching the starter (score 29), and the crib must be 4466 (score 24), or vice versa.
• The highest number of points possible (excluding pegging points) in one round is 77. The dealer must score 53, the opponent must then have the other 4466 making another 24 point hand for a total of 77.
• The highest number of points from a hand that has a potential to be a "19 hand" is 15. It is a crib hand of one suit, 46J and another ten card, with a 5 of that suit cut up. The points are 15 for 6, a run for 9, nobs for 10, and a flush for 15. Any of the following cards in an unlike suit yields a "19 hand"; 2,3,7,8,and an unpaired ten card.
• The most points that can be pegged by playing one card is 15, by completing a double pair royal on the last card and making the count 15: 12 for double pair royal, 2 for the 15, and 1 for the last card. This can happen in two ways in a two-player game. The non-dealer must have two ten-value cards and two 2s, and the dealer must have one ten-value card and 722, in which case the play must go: 10-10-10-go; 7-2-2-2-2. For example:

Alice (dealer)
Bob

Player	Card	Cumulative	Score	Announced
Bob		10		"ten"
Alice		20		"twenty"
Bob		30	3 points (run)	"thirty"
Alice			1 point to Bob (30 for one)	"go"
Alice		7		"seven"
Bob		9		"nine"
Alice		11	2 points	"eleven for two"
Bob		13	6 points	"thirteen for six"
Alice		15	15 points (double pair royal, fifteen, last card)	"fifteen for fifteen"

• Alternatively, the players can each have two deuces, with one also holding A-4 and the other two aces. Then play might go 4-A-A-A-2-2-2-2.
• The maximum number of points that can be scored in a single deal by the dealer in a two player game is 78 (pegging + hand + crib):
  Non-dealer is dealt 3 3 4 4 5 J and Dealer is dealt 3 3 4 4 5 5. Non-dealer discards J 5 to the crib (as ill-advised as this may be). Dealer discards 5 5 to the crib. Note that the J is suited to the remaining 5. The remaining 5 is cut.
  Play is 3 3 3 3 4 4 4 4 go. The dealer scores 29 total peg points.
  The dealer's hand is 3 3 4 4 5 = 20
  The dealer's crib is J(nobs) 5 5 5 5 = 29
  The total score for the dealer is 29 + 20 + 29 = 78.
  Note that the correct play for both players is to keep 3 3 4 5 worth 10 points and discarding J 4 and 4 5 to the crib respectively, meaning in reality, this hand would never take place. A more realistic hand would be both players being dealt 3 3 4 4 J J with both discarding J J and a 5 cut. In this case, with pegging as described above, the total score would be 20 (hand) + 21 (crib) + 29 (pegging) = 70 points.
• The maximum number of points that can be scored in a single deal by the non-dealer in a two player game is 48 (pegging + hand), with the following example :
  Non-dealer is dealt 5 5 4 4 crib crib and Dealer is dealt 4 4 5 9 crib crib. Cut card is a 6.
  Play is 5 5 5 4 4 4 4, with the Non-dealer pegging 24. The Non-dealer scores 24 in the hand for a total of 48 points.
• The maximum number of points that can be scored with a four-card flush is 21, which is achieved with a hand of 5 5 10 J Q or 5 5 J Q K: a pair, six fifteens, a three-card sequence, and the flush. A five-card flush of 5 10 J Q K scores 18 if the Jack is not the starter.

Minimum scores

• The dealer in two-player, 6-card cribbage will always peg at least one point during the play (the pegging round), unless the opponent wins the game before the pegging is finished. If non-dealer is able to play at each turn then dealer must score at least one for "last"; if not, then dealer scores at least one for "go".
• While 19 is generally recognized as "the impossible hand", meaning that there is no combination of 5 cards that will produce a score of 19 points, scores of 25, 26, 27, and greater than 29 are also impossible in-hand point totals.[1] Sometimes if a player scores 0 points in their hand they will claim they have a "19-point hand."[3]

Minimum while holding a 5

If a player holds a 5 in their hand, that player is guaranteed at least two points, as shown below:

A 0-point hand must have five distinct cards without forming a run or a fifteen combination. If such a hand includes a 5, it cannot hold a 10 or a face card. It also cannot include both an A and a 9; both a 2 and an 8; both a 3 and a 7; or both a 4 and a 6. Since four more cards are needed, exactly one must be taken from each of those sets. Let us run through the possible choices:

• If the hand includes a 9, it cannot hold a 6, so it must hold a 4. Having both a 4 and a 9, it cannot hold a 2, so it must hold an 8. Holding both a 4 and an 8, it cannot hold a 3, so it must hold a 7. But now the hand includes a 7-8 fifteen, which is a contradiction.
• Therefore, the hand must include an A. If the hand includes a 7, it now cannot contain an 8, as that would form a 7-8 fifteen. However it cannot hold a 2, as that would form a 7-5-2-A fifteen. This is a contradiction.
• Therefore, the hand must include a 3. Either a 2 or a 4 would complete a run, so the hand must therefore include a 6 and an 8. But this now forms an 8-6-A fifteen, which is a contradiction.

Therefore, every set of five cards including a 5 has a pair, a run, or a fifteen, and thus at least two points.

Interestingly, a hand with two 5s also can score only two points; an example is 2 5 5 7 9, which would be most likely a crib hand, and would not score a flush because of the pair, although said hand can be a non-crib four-card flush if either 5 is the starter. A hand with three 5s scores at least eight points; a hand with all four 5s scores 20 points and is improved only with a 10, J, Q, or K (scoring 28 except for the 29 hand previously described.)

It is also true that holding both a 2 and a 3, or an A and a 4 (pairs of cards adding up to five) also guarantees a non-zero score:

• If a hand includes both a 2 and a 3 and is to score 0 points, it cannot have a face card, an A, a 4, or a 5. This requires three cards from the 6, 7, 8, and 9, and any such selection will include a fifteen.
• If a hand includes both an A and a 4 and is to score 0 points, it cannot have a face card or a 5. It also cannot have both a 2 and a 3; both a 6 and a 9; or both a 7 and an 8. If the hand includes a 2, it cannot have a 9 (9-4-2 fifteen). Thus it must have a 6. It then cannot have an 8 (8-4-2-A fifteen) or a 7 (7-6-2 fifteen). If, however, the hand includes a 3, it cannot include an 8 (8-4-3 fifteen) or a 7 (7-4-3-A fifteen). These are all contradictions, so every hand containing both an A and a 4 scores at least two points.

Odds

• The table below assumes the card(s) discarded to the crib are randomly chosen. Given this assumption, the odds of getting a 28 hand in a two-player game are about 1 in 170984, and a perfect 29 hand 1 in 3,248,700.[3]

• However, if we assume that the player will always keep J555 if those cards are included in the hand, the odds of getting a perfect 29 hand starting with a six-card hand are 1 in 216,580, while the odds after discarding from a five-card hand are 1 in 649,740.[4]

Scoring Breakdown, assuming random discard(s) to the crib[1]

Score	Number of hands (out of 12,994,800)	Percentage of hands	Percentage of hands at least as high
0	1009008	7.7647	100
1	99792	0.7679	92.2353
2	2813796	21.6532	91.4674
3	505008	3.8862	69.8142
4	2855676	21.9755	65.928
5	697508	5.3676	43.9525
6	1800268	13.8538	38.5849
7	751324	5.7817	24.7311
8	1137236	8.7515	18.9494
9	361224	2.7798	10.1979
10	388740	2.9915	7.4181
11	51680	0.3977	4.4266
12	317340	2.4421	4.0289
13	19656	0.1513	1.5868
14	90100	0.6934	1.4355
15	9168	0.0706	0.7421
16	58248	0.4482	0.6715
17	11196	0.0862	0.2233
18	2708	0.0208	0.1371
19	0	0	0.1163
20	8068	0.0621	0.1163
21	2496	0.0192	0.0542
22	444	0.0034	0.0350
23	356	0.0027	0.0316
24	3680	0.0283	0.0289
25	0	0	0.0006
26	0	0	0.0006
27	0	0	0.0006
28	76	0.0006	0.0006
29	4	0.00003	0.00003

• Mean = 4.7692
• Standard deviation = 3.1254
• Skewness = 0.9039
• Excess kurtosis = 1.4599

Note that these statistics do not reflect frequency of occurrence in 5 or 6-card play. For 6-card play the mean for non-dealer is 7.8580 with standard deviation 3.7996, and for dealer is 7.7981 and 3.9082 respectively. The means are higher because the player can choose those four cards that maximize their point holdings. For 5-card play the mean is about 5.4.

Slightly different scoring rules apply in the crib - only 5-point flushes are counted, in other words you need to flush all cards including the turn-up and not just the cards in the crib. Because of this, a slightly different distribution is observed:

Scoring Breakdown (crib/box hands only)

Score	Number of hands (+/- change from non-crib distribution) (out of 12,994,800)	Percentage of hands	Percentage of hands at least as high
0	1022208 (+13200)	7.8663	100
1	99792 (0)	0.7679	92.1337
2	2839800 (+26004)	21.8534	91.3658
3	508908 (+3900)	3.9162	69.5124
4	2868960 (+13284)	22.0778	65.5962
5	703496 (+5988)	5.4137	43.5184
6	1787176 (-13092)	13.7530	38.1047
7	755320 (+3996)	5.8125	24.3517
8	1118336 (-18900)	8.6060	18.5393
9	358368 (-2856)	2.7578	9.9332
10	378240 (-10500)	2.9107	7.1755
11	43880 (-7800)	0.3377	4.2648
12	310956 (-6384)	2.3929	3.9271
13	16548 (-3108)	0.1273	1.5342
14	88132 (-1968)	0.6782	1.4068
15	9072 (-96)	0.0698	0.7286
16	57288 (-960)	0.4409	0.6588
17	11196 (0)	0.0862	0.2179
18	2264 (-444)	0.0174	0.1318
19	0 (0)	0	0.1144
20	7828 (-240)	0.0602	0.1144
21	2472 (-24)	0.0190	0.0541
22	444 (0)	0.0034	0.0351
23	356 (0)	0.0027	0.0317
24	3680 (0)	0.0283	0.0289
25	0 (0)	0	0.0006
26	0 (0)	0	0.0006
27	0 (0)	0	0.0006
28	76 (0)	0.0006	0.0006
29	4 (0)	0.00003	0.00003

• Mean = 4.7348

As above, these statistics do not reflect the true distributions in 5 or 6 card play, since both the dealer and non-dealer will discard tactically in order to maximise or minimise the possible score in the crib/box.

Card combinations

• A hand of four aces (AAAA) is the only combination of cards wherein no flip card will add points to its score.
• There are 71 distinct combinations of card values that add to 15:

Two cards	Three cards			Four cards				Five cards
X5 96 87	X4A X32 95A 942 933	86A 852 843 77A 762	753 744 663 654 555	X3AA X22A 94AA 932A 9222 85AA	842A 833A 8322 76AA 752A 743A	7422 7332 662A 653A 6522 644A	6432 6333 554A 5532 5442 5433 4443	X2AAA 93AAA 922AA 84AAA 832AA 8222A 75AAA	742AA 733AA 7322A 72222 66AAA 652AA 643AA	6422A 6332A 63222 553AA 5522A 544AA 5432A	54222 5333A 53322 4442A 4433A 44322 43332
Note: "X" indicates a card scoring ten: 10, J, Q or K

Hand and Crib statistics

If both the hand and the crib are considered as a sum (and both are drawn at random, rather than formed with strategy as is realistic in an actual game setting) there are 2,317,817,502,000 (2.3 trillion) 9-card combinations. ${\displaystyle {52 \choose 4}\times {48 \choose 4}\times 44=2,317,817,502,000}$

• As stated above, the highest score a dealer can get with both hand and crib considered is 53.
• The only point total between 0 and 53 that is not possible is 51.

Scoring Breakdown

Score	Number of hand-crib pairs (out of 2,317,817,502,000)	Percentage of hand-crib pairs to 6 decimal places	Percentage of hand-crib pairs at least as high
0	14485964652	0.624983	100
1	3051673908	0.131662	99.375017
2	80817415668	3.486789	99.243356
3	23841719688	1.028628	95.756566
4	190673505252	8.226424	94.727938
5	70259798952	3.031291	86.501514
6	272593879188	11.7608	83.470222
7	121216281624	5.22976	71.709422
8	290363331432	12.527446	66.479663
9	151373250780	6.530853	53.952217
10	254052348948	10.960843	47.421364
11	141184445960	6.091267	36.460521
12	189253151324	8.165145	30.369254
13	98997926340	4.27117	22.204109
14	127164095564	5.486372	17.932939
15	59538803512	2.568744	12.446567
16	77975659056	3.364185	9.877823
17	32518272336	1.402969	6.513638
18	42557293000	1.836093	5.110669
19	17654681828	0.761694	3.274576
20	22185433540	0.957169	2.512881
21	8921801484	0.384923	1.555712
22	10221882860	0.441013	1.17079
23	4016457976	0.173286	0.729776
24	5274255192	0.227553	0.55649
25	1810154696	0.078097	0.328938
26	2305738180	0.099479	0.25084
27	750132024	0.032364	0.151361
28	1215878408	0.052458	0.118998
29	401018276	0.017302	0.06654
30	475531940	0.020516	0.049238
31	184802724	0.007973	0.028722
32	233229784	0.010062	0.020749
33	82033028	0.003539	0.010686
34	71371352	0.003079	0.007147
35	19022588	0.000821	0.004068
36	44459120	0.001918	0.003247
37	9562040	0.000413	0.001329
38	10129244	0.000437	0.000916
39	1633612	0.00007	0.000479
40	5976164	0.000258	0.000409
41	1517428	0.000065	0.000151
42	600992	0.000026	0.000085
43	127616	0.000006	0.00006
44	832724	0.000036	0.000054
45	222220	0.00001	0.000018
46	42560	0.000002	0.000009
47	24352	0.000001	0.000007
48	119704	0.000005	0.000006
49	6168	0	0
50	384	0	0
51	0	0	0
52	4320	0	0
53	288	0	0

• Mean: 9.50397
• Median: 9
• Mode: 8

See also

• Cribbage (rules)
• Cribbage (strategy)

References

1. Steven S. Lumetta (2007-05-15). "Amusing Cribbage Facts". Archived from the original on 2018-01-16. Retrieved 2008-03-03.
2. Tim Wood (2008-08-05). "All Possible Cribbage Hands". Archived from the original on 2013-02-09. Retrieved 2008-08-05.
3. Weisstein, Eric W. "Cribbage". MathWorld. Retrieved 2008-03-02. All scores from 0 to 29 are possible, with the exception of 19, 25, 26, and 27. For this reason, hand scoring zero points is sometimes humorously referred to as a "19-point" hand.
4. Cribbage Corner (2008-05-05). "Perfect cribbage hand odds". Retrieved 2008-05-05.