# Cribbage statistics

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

## 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.
• 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:
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 & 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.

## 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 five

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 any face cards. 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 5 cards including a 5 has a pair, a run, or a fifteen, and thus at least two points.

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 odds of getting a 28 hand in a two-player game are 1 in 15,028.
• The odds of getting a perfect 29 hand in a two-player game are 1 in 216,580.[4]
• The odds of getting a perfect 29 hand in a three- or four-player game are 1 in 649,740.

Scoring Breakdown[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

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

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