Poker Test
Every draw is a hand of five. Sort each Powerball draw by how its numbers' last digits cluster — matching pairs, three of a kind, full house, four of a kind. Compare what the lottery actually dealt to what the math says it should.
| Hand | Expected | Observed | Bars | z-score |
|---|---|---|---|---|
| High card | 34.99% ≈ 468.6 | 33.53% 449 hands | -1.12 | |
| One pair | 49.99% ≈ 669.4 | 49.66% 665 hands | -0.24 | |
| Two pair | 9.18% ≈ 122.9 | 11.43% 153 hands | +2.84 | |
| Three of a kind | 5.10% ≈ 68.3 | 4.93% 66 hands | -0.29 | |
| Full house | 0.55% ≈ 7.3 | 0.30% 4 hands | -1.23 | |
| 4 of a kind | 0.18% ≈ 2.4 | 0.15% 2 hands | -0.28 | |
| 5 of a kind | 0.00% ≈ 0.0 | 0.00% 0 hands | — |
4-of-a-kind and 5-of-a-kind hands are so rare (roughly 1 in 550 and 1 in 57,000 respectively) that observing zero at this sample size is expected — their z-scores would be meaningless and are suppressed when expected count is below one.
How it works
Grouping by last digit. Each white ball has a last digit 0–9. Two balls share a “pair” if their last digits match (e.g., 7 and 17). A “three of a kind” is three last digits matching (e.g., 3, 23, 43). This turns every five-number draw into a seven-category hand, exactly like poker.
Expected frequencies. Computed by enumeration over all C(69, 5) = 11,238,513 possible hands. Last digit 0 has only 6 members in 1–69 (10, 20, 30, 40, 50, 60) while the other nine digits have 7 each, so probabilities are computed from that exact distribution rather than assuming uniform bucket sizes.
The test. The table shows how many draws landed in each hand type (observed) alongside the count predicted by combinatorics (expected), together with a z-score for each row and an overall chi-square goodness-of-fit. If Powerball is fair, the distribution across the seven hand types should match the math up to sampling noise.