Sets, runs and what this article assumes
Rummy-style games, including gin rummy and its relatives, ask players to arrange cards into melds. A set (sometimes called a book or triple) is three or four cards of the same rank, such as three sevens. A run (or sequence) is three or more consecutive cards of the same suit, such as the 4, 5 and 6 of clubs.
Rules vary a good deal between games, so the calculations here use a simple, clearly stated setup: one standard 52-card deck, no jokers, aces counted low only (so A-2-3 is a run but Q-K-A is not), and melds of at least three cards. Many popular 13-card variants use several decks and jokers, so the figures below illustrate the arithmetic and should not be read as the odds in any specific game.
Starting with two cards
The simplest cases are two-card combinations. There are 1,326 different pairs of cards in a 52-card deck (52 x 51 / 2).
- A pair of the same rank. There are 13 ranks, and each has 6 ways to choose two of its four cards, so 78 pairs. The probability is 78/1326 = 3/51, about 5.9%. That matches the direct reasoning: whatever your first card is, three of the remaining 51 cards share its rank.
- Two adjacent cards of the same suit. In each suit there are 12 neighbouring rank pairs (A-2 up to Q-K), so 48 across four suits. That gives 48/1326, about 3.6%.
So a same-rank pair is somewhat more likely than a connected same-suit pair, which is the first hint that sets tend to turn up a bit more often than runs.
Three-card melds
How many three-card combinations exist? There are 52 x 51 x 50 / 6 = 22,100. Now count the melds:
- Sets: 13 ranks x 4 ways to choose three of four suits = 52 sets.
- Runs: with ace low and no wrap-around, a three-card run in a suit can start at any rank from ace up to jack (A-2-3 through J-Q-K), which is 11 starting ranks. So 11 per suit, 44 in total.
The chance that three random cards form a set is 52/22,100, about 0.24%, or 1 in 425. The chance that they form a run is 44/22,100, about 0.20%, or roughly 1 in 502. Together, 96/22,100 is about 0.43%.
Most dealt triples are not melds. The play of the game is about assembling them gradually, not being dealt them whole.
Larger hands
With more cards, many more triples become possible at once, and the chance of at least one meld grows quickly. The figures below were found by exact counting over all possible hands (not by simulation) under the assumptions above, with hand sizes chosen because they are common starting hands in different games:
| Cards dealt | At least one set | At least one set or run |
|---|---|---|
| 7 | about 7.7% | about 13.8% |
| 10 | about 24.1% | about 39.8% |
| 13 | about 49.0% | about 70.7% |
Two patterns stand out. First, the chance rises steeply: going from 7 to 13 cards multiplies the opportunity for a meld because the number of three-card subsets of the hand grows from 35 to 286. Second, adding runs to sets lifts the probability by a large share, which shows that runs matter in practice even though a single triple is a little less likely to be a run than a set.
The pair figures behave similarly. The chance that a 7-card hand contains at least one pair or better is about 79.0%, and for 10 cards it is about 98.1%. A hand with no two cards of the same rank is therefore rare once you have more than a handful of cards.
Why sets and runs differ in practice
The headline numbers compare one meld type with another, but a few features change how the game plays out:
- Flexibility. A pair needs one of two particular cards to become a set. A pair of connected cards in the middle of a suit, such as 7 and 8, has two cards that extend it (6 and 9), while one at the end of a suit (A-2 or Q-K) has only one. Middle connectors are therefore a little more flexible.
- Cards in play. Each time a card from a potential meld is discarded or taken, the outs shrink. The idea of counting outs applies directly, and so does the effect of drawing without replacement.
- Overlap. One card can often belong to either a set or a run, but it can only be used in one meld at a time. The 6 of hearts might sit with two other sixes or with the 5 and 7 of hearts, though not with both groups together.
Reading the numbers carefully
A few cautions apply to all of this:
- The table is about the dealt hand. Real games let you draw and discard, so the chance of ending with melds is higher than the initial deal suggests, and depends on strategy and on opponents.
- Having a meld does not decide a hand. Deadwood, scoring rules and going out all matter.
- The figures assume a freshly shuffled deck. Past deals do not change them.
- The counting uses combinations, so more detail on the method can be found under permutations combinations.
- Where cards are played for money, the usual caution applies: games with a built-in disadvantage lose over time, outcomes are uncertain, and nobody should play with money they cannot afford to lose.
Checking the arithmetic yourself
You can verify the three-card results by hand. Take 52 sets and 44 runs out of 22,100 triples: 52 + 44 = 96, and 96/22,100 is 0.0043, or 0.43%. For pairs, check that 78 x 17 = 1,326, which confirms that 78/1326 is exactly 1/17. Small checks like these catch most errors before they reach a table, and they are a reminder that a single deal says little compared with the long-run proportions.