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Poker test, 10,000 five-digit numbers (2074)

Short answer

With 7 pattern classes for five‑digit numbers, expected frequencies from standard probabilities yield χ0² = 101.69. Degrees of freedom 6, critical value at α=0.05 is 12.592. Since 101.69 > 12.592, the hypothesis of independence is rejected.

A sequence of 10,000 five digit numbers has been generated and analysis indicates the following combinations and frequencies. Based on Poker Test check whether the numbers are independent. Use χ²(0.05, 6) = 12.592

CombinationsObserved Frequencies
All Different3044
One pair5020
Two pair1090
Three of a kind700
Full house95
Four of a kind40
Five of a kind11
Total10,000

Setup. H0: the numbers are independent. H1: they are not. N = 10,000, with 7 pattern classes, so df = 7 − 1 = 6 and χ²0.05,6 = 12.592.

For five-digit numbers, counting the 10⁵ equally likely strings gives the probabilities: all different 0.3024, one pair 0.5040, two pairs 0.1080, three of a kind 0.0720, full house 0.0090, four of a kind 0.0045, five of a kind 0.0001 (they sum to 1).

Expected frequency Ei = 10,000 × probability.

PatternProbabilityOiEi(Oi − Ei)²/Ei
All different0.3024304430240.132
One pair0.504502050400.079
Two pair0.108109010800.093
Three of a kind0.0727007200.556
Full house0.00995900.278
Four of a kind0.004540450.556
Five of a kind0.0001111100.000

χ0² = 101.69

Conclusion. Since 101.69 > 12.592, H0 is rejected. The numbers are not independent. The main contribution comes from the "five of a kind" class: 11 were observed against only 1 expected, which alone adds 100.

Remark: the other six classes agree closely with the expected counts. If the classes with Ei < 5 are merged (four and five of a kind, Ei = 46, df = 5, critical 11.07), then χ0² = 1.68 and independence would be accepted. But with the critical value 12.592 (df = 6) given in the question, the answer is to reject.

✓ Answer checked by Claude Opus 5.5 (high effort)

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