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Poker test, 1000 four-digit numbers (2073)

Short answer

Using probabilities for four‑digit numbers (all different 0.504, one pair 0.432, two pairs 0.027, three like 0.036, four like 0.001), χ0² = 33.74. With df=4, critical value at α=0.05 is 9.488. Since 33.74 > 9.488, independence is rejected.

A sequence of 1000 four digit numbers has been generated and an analysis indicates the following combination and frequencies:

Combination (i)Observed Frequency (Oi)
Four different digit520
One pair390
Two pair55
Three like digit34
Four like digit1
1000

Setup. H0: the numbers are independent. H1: they are not. N = 1000, five patterns for four-digit numbers, so df = 5 − 1 = 4. The question gives no α; take α = 0.05, so χ²0.05,4 = 9.488.

For four-digit numbers the probabilities are: four different 0.504, one pair 0.432, two pairs 0.027, three of a kind 0.036, four of a kind 0.001.

PatternProbabilityOiEi(Oi − Ei)²/Ei
Four different0.5045205040.508
One pair0.4323904324.083
Two pairs0.027552729.037
Three of a kind0.03634360.111
Four of a kind0.001110.000

χ0² = 33.74

Conclusion. Since 33.74 > 9.488, H0 is rejected; the numbers are not independent. The main cause is "two pairs": 55 observed against 27 expected (contribution 29.04). Merging the last two classes (Ei = 37, df = 3, critical 7.815) gives χ0² = 33.74 and the same decision.

✓ Answer checked by Claude Opus 5.5 (high effort)

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