Gap test algorithm; formulate poker test

Algorithm for the gap test

The gap test checks independence by examining the gaps between successive occurrences of a chosen digit. For a single digit the chance of occurrence is p = 0.1 (for an interval [α, β), p = β − α). Gap length x has probability p(1 − p)^x and cdf F(x) = 1 − (1 − p)^(x+1).

1. Choose the target digit or interval [α, β).
2. Scan the sequence and note each position where the target occurs. The gap is the count of non-target numbers between two consecutive occurrences.
3. Group the n gaps by length (0, 1, 2, ..., and a last class "x or more").
4. Find the observed relative frequencies and the cumulative function SN(x).
5. Compute the theoretical F(x) = 1 − (1 − p)^(x+1).
6. Compute D = max|F(x) − SN(x)| (K-S) and compare it with Dα for n gaps. Alternatively use χ0² = Σ (Oi − Ei)²/Ei with Ei = n·p(1 − p)^x.
7. If the statistic is greater than the critical value, reject independence; else accept.

Four-digit poker test

The poker test treats each group of four digits like a poker hand and compares the observed counts of each pattern with the expected counts, using the chi-square test. Each random number is taken to four decimal places (e.g. 0.5127 gives digits 5, 1, 2, 7).

Probabilities of the patterns

Probabilities are found by counting: all different = 10·9·8·7/10⁴ = 0.504; four alike = 10/10⁴ = 0.001; and so on. They sum to 1.

Steps: (1) Take N numbers and classify each into a pattern. (2) Expected count Ei = N × probability. (3) Compute χ0² = Σ (Oi − Ei)²/Ei. (4) Degrees of freedom = classes − 1 = 4. (5) Reject independence if χ0² > χ²α,4.

Example. Suppose N = 1000 four-digit numbers give these counts (data assumed for illustration):

χ0² = 1.069. With α = 0.05 and df = 4, the critical value is 9.488. Since 1.069 < 9.488, we do not reject H0, and the numbers are accepted as independent.