Exact category counts
Assumptions: 52 distinct cards, no jokers, three cards dealt without replacement, 12 recognised sequence patterns including A‑2‑3 and Q‑K‑A, and each hand assigned to its highest qualifying category.
| Category | Combinations | Probability |
|---|---|---|
| Trail | 52 | 0.2353% |
| Pure sequence | 48 | 0.2172% |
| Sequence (not pure) | 720 | 3.2579% |
| Colour (not pure sequence) | 1,096 | 4.9593% |
| Pair | 3,744 | 16.9412% |
| High card | 16,440 | 74.3891% |
| Total | 22,100 | 100% (before rounding) |
Notice that pure sequences are slightly rarer than trails under these assumptions, yet trails rank higher in the classic game. Ranking is a rule convention; it is not simply an automatic ordering by rarity.
How the numbers are obtained
The total is 52 × 51 × 50 ÷ 6 = 22,100. Divide by six because the same three cards can be dealt in six orders.
- Trail: choose one of 13 ranks and three of its four suits:
13 × 4 = 52. - Pure sequence: choose one of 12 rank patterns and one of four suits:
12 × 4 = 48. - Mixed sequence: each pattern has
4³ = 64suit assignments. Remove the four same-suit assignments:12 × 60 = 720. - Colour: choose a suit and any three of its 13 ranks, then remove pure sequences:
4 × 286 − 48 = 1,096. - Pair: choose the pair rank, two of its four suits, a different spare rank and that card’s suit:
13 × 6 × 12 × 4 = 3,744. - High card: subtract the five preceding exclusive categories from the total:
22,100 − 5,660 = 16,440.
What changes if A‑2‑3 moves?
Moving A‑2‑3 from highest to second or lowest changes comparisons between runs. It does not change these category counts, provided A‑2‑3 still qualifies as a sequence. Excluding it entirely, adding jokers, or using a variant with wild cards requires a different count.
The sequence guide explains the ordering conventions. These counts describe the classic deck model, not an audit of any app’s random-number generator.
Rarity is not your chance of winning a live round
A category probability describes a fresh random three-card hand under the assumptions above. A round’s outcome also depends on other hands, the number of players, rule choices and decisions to leave the round. Information seen during play changes the relevant question.
No sequence of previous losses makes a strong hand “due”. These figures do not establish an earning strategy, a predictable return or the fairness of an online platform.
Reproduce the result
The delivery package includes tools/verify_math.py, which enumerates every unordered three-card combination and checks the category totals independently of this table. You can also reproduce the formulas with a calculator.
Use the result for learning: compare how often a category occurs and why category membership must be checked before high-card comparisons.