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Cribbage Hand Odds: Every Score, and How Often It Happens

We scored all 12,994,800 possible cribbage hands — every four-card hand against every starter it could meet — and counted the results. This is the complete distribution, and it settles a few arguments.

The numbers that matter

Two of those are worth pausing on. The mean cribbage hand is worth 4.77 points — not the 6 or 8 most players guess, because the handful of enormous hands everyone remembers are drowned by how ordinary the rest are. And one hand in thirteen scores nothing at all. If you feel like you are being dealt rubbish, you are: it happens about eight times in a hundred, and over a typical ten-deal game there is a better-than-even chance of at least one.

The full distribution, 0 to 29

Every score, how many of the 12,994,800 combinations produce it, and the odds. Four scores are missing entirely, which is not an error — see below.

Percentages are rounded to four decimal places; the 29 row is 0.0000308%.

The shape of the curve

Laid out as a chart, the distribution has a character that a table hides: it is not a smooth bell. It is a comb.

Scores 25, 26 and 27 are omitted — like 19, they cannot occur. Gold bars mark scores that turn up less than once in two hundred hands.

The teeth are the story. Even scores tower over their odd neighbours all the way up, and the gaps at 11, 13, 15 and 17 are deep enough to see from across the room. The other feature worth noticing is the small hump at 24 — a score that is more likely than 22 or 23, against the general downward trend. That is not noise. Twenty-four is the natural ceiling of several very common shapes at once (double-double runs with fifteens, and four cards around a 5), so a cluster of otherwise unrelated hands all land on the same number.

Why four out of five hands are even

80.7% of cribbage hands score an even number and only 19.3% score an odd one. Once you see why, you cannot unsee it, and it is genuinely useful at the table.

Almost everything in cribbage scores in twos. Every fifteen is 2. Every pair is 2. Those two categories account for the overwhelming majority of all points scored in the game, and neither can ever produce an odd total. Only three things can make a hand odd:

• A run of odd length — a run of three scores 3, a run of five scores 5. (A run of four scores 4, and a double run of three scores 6, so those keep you even.)
• A flush — 5 for a five-card flush, or 4 in hand, so only the five-card version flips the parity.
• His nobs — the single most reliable source of odd numbers in the game, and worth exactly 1.

The four impossible scores

19, 25, 26 and 27 never appear. Not "rarely" — the sweep found exactly zero of each across all 12,994,800 combinations. They are arithmetically unreachable.

The reason is that hand scores are not free integers; they are sums drawn from a restricted menu, and above about 18 the menu gets very thin. To score in the twenties you need a hand built around multiple fifteens and multiple pairs or a multiple run, and those structures land on 20, 21, 22, 23, 24, 28 and 29 — there is simply no arrangement of five cards whose components add to 25, 26 or 27. Nineteen fails for the same reason lower down.

Because 19 is impossible, "a nineteen hand" became cribbage slang for a hand that scores nothing — a joke about claiming an impossible number. It is the one piece of cribbage vocabulary that every player knows and no rulebook contains. There is more on all four, and on the 29, in the 29 hand and other scoring oddities.

What "an average hand" really means

The mean of 4.77 is the average over every hand you could hold, which is not the average of the hands you will actually count in a game — and the difference matters.

In a real deal you are dealt six cards and keep the best four. That choice is worth several points on its own: you are not sampling the distribution above at random, you are taking the best of fifteen draws from it. A competent player's counted hand averages closer to eight, and their crib rather less. So 4.77 is not a target — it is the baseline you beat by discarding well, and the gap between 4.77 and what you actually count is a fair measure of how well you are choosing.

Which is why the discard is the most important decision in the game, and why we worked out the expected value of every four-card keep as a separate exercise.

Making this useful at the table

Four things worth carrying away from the table above:

• Twelve is a good hand. Only 4% of hands reach it. Players routinely talk themselves out of a certain 12 while chasing a speculative 16 that arrives one time in a hundred and fifty.
• Zero is normal. Nearly 8% of hands, and around 30% score two or less. A bad hand is not bad luck to be complained about; it is a scheduled event you should be discarding defensively for when the crib is your opponent's.
• Odd totals need a reason. See the parity check above. It is the cheapest error-catcher in cribbage.
• Chasing 24 is a bad plan. It is 1 in 3,531. The points you give up holding out for it will cost you far more over a game than the hand would ever have paid.

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