The complete standard sample space
C(52,3) = 22,100 unordered hands. The six category counts sum exactly to that total under the stated assumptions.
Assumptions and sample space
Use one standard 52-card deck, three cards without replacement, no jokers and twelve valid rank runs including A-2-3 and Q-K-A. Treat pure sequences separately from ordinary sequences and exclude runs from colour. Ordering A23 above or below AKQ changes comparisons, not these category counts.
C(52,3) = 52 × 51 × 50 ÷ 6 = 22,100. Each unordered hand has the same probability under a fair uniform deal. Multiple opponents, visible cards, wild cards and nonstandard dealing require a different calculation.
Category counts
| Category | Hands | Probability |
|---|---|---|
| Trail | 52 | 0.2353% |
| Pure sequence | 48 | 0.2172% |
| Sequence, excluding pure | 720 | 3.2579% |
| Colour, excluding pure | 1,096 | 4.9593% |
| Pair | 3,744 | 16.9412% |
| High card | 16,440 | 74.3891% |
| Total | 22,100 | 100% (before rounding) |
Rounding individual percentages can produce a tiny displayed total difference. The integer counts sum exactly to the full sample space.
Where the numbers come from
- Trail: 13 ranks × C(4,3) suit selections = 52.
- Pure sequence: 12 rank runs × 4 suits = 48.
- Ordinary sequence: 12 × (4³ − 4) suit assignments = 720.
- Colour: 4 × (C(13,3) − 12) = 1,096.
- Pair: 13 × C(4,2) × 12 × 4 = 3,744.
- High card: subtract the other five categories from 22,100 = 16,440.
Rarity is not a winning percentage
A pure sequence is rarer than a trail under these assumptions, yet the standard game ranks a trail above it. The rulebook sets category order. Similarly, the chance of being dealt a pair is not the chance that a particular pair beats all remaining hands.
To calculate showdown outcomes, you would need the number of opponents, blocked cards, the exact comparison rules and the selection process that brought those hands to the show. A category frequency alone does not justify a deposit, a stake size or an earnings forecast.
Try a testable learning question
Why is a colour more common than a pure sequence? There are many three-rank selections within a suit but only twelve permitted consecutive sets. Draw a few examples, then return to the ranking chart to keep probability and game hierarchy separate.
Reproducible calculation method
Enumerate every three-element subset of a 52-card deck. Classify each subset once: trail first, then pure sequence, sequence, colour, pair and high card. This ordering keeps overlapping descriptions from being counted twice. The 22,100 subsets split into the six counts shown above.
The hand-comparison algorithm is checked against all of these combinations. For the exact assumptions and the distinction between game math and APK tests, see the source and methodology register.