Cribbage statistics

Main article: Cribbage

Some cribbage statistics are

Distinct hands

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

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

Maximum scores

Alice
(dealer)
Bob
Player Card Cumulative Score Announced
Bob10"ten"
Alice20"twenty"
Bob303 points (run)"thirty"
Alice1 point to Bob (30 for one)"go"
Alice7"seven"
Bob9"nine"
Alice112 points"eleven for two"
Bob136 points"thirteen for six"
Alice1515 points (double pair royal,
fifteen, last card)
"fifteen for fifteen"

Minimum scores

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:

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:

Odds


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

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. {52 \choose 4} \times {48 \choose 4} \times 44 = 2,317,817,502,000

Scoring Breakdown

Score Number of hand-crib pairs
(out of 2,317,817,502,000)
Percentage of hand-crib pairs 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

See also

References

  1. 1 2 3 Steven S. Lumetta (2007-05-15). "Amusing Cribbage Facts". Retrieved 2008-03-03.
  2. Tim Wood (2008-08-05). "All Possible Cribbage Hands". 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.
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