• B.2 Expected Difference in Proportions The exact formula for the expected difference between the relative frequency of 1 for k/1-streak successor trials and the relative frequency of 1 for k/0-streak successor trials can be obtained with an approach similar to the previous section. The difference satisfies Dk := P1k (1 P0k ), − − and there are three categories of sequences for which Dk is defined: (1) a sequence that ends in a run of 0s of length k or more with f0k 2 and f1k 1, with the difference equal to D1 = ≥ ≥ (f1k s1k )/f1k (s0k 1)/(f0k 1), (2) a sequence that ends in a run of 1s of length k or more − − − − with f0k 1 and f1k 2, with the difference equal to D2 := (f1k s1k )/(f1k 1) s0k /f0k (3) a ≥ ≥ − − − sequence that ends in a run of 0s of length k 1 or less or a run of 1s of length k 1 or less, − − with f0k 1 and f1k 1, with the difference equal to D3 := (f1k s1k )/f1k s0k /f0k . For all other ≥ ≥ − − sequences the difference is undefined. Theorem 8 For n > 0 and n1, n0, k with n0 + n1 = n and 1 < k n0, n1 n, the expected ≤ ≤ difference in the relative frequency of 1 for k/1-streak successors that are 1 and the relative frequency
• B.2 Expected Difference in Proportions The exact formula for the expected difference between the relative frequency of 1 for k/1-streak successor trials and the relative frequency of 1 for k/0-streak successor trials can be obtained with an approach similar to the previous section. The difference satisfies Dk := P1k − (1 − P0k ), and there are three categories of sequences for which Dk is defined: (1) a sequence that ends in a run of 0s of length k or more with f0k ≥ 2 and f1k ≥ 1, with the difference equal to D1 = (f1k − s1k )/f1k − (s0k − 1)/(f0k − 1), (2) a sequence that ends in a run of 1s of length k or more with f0k ≥ 1 and f1k ≥ 2, with the difference equal to D2 := (f1k − s1k )/(f1k − 1) − s0k /f0k (3) a sequence that ends in a run of 0s of length k − 1 or less or a run of 1s of length k − 1 or less, with f0k ≥ 1 and f1k ≥ 1, with the difference equal to D3 := (f1k − s1k )/f1k − s0k /f0k . For all other sequences the difference is undefined. Theorem 8 For n > 0 and n1, n0, k with n0 + n1 = n and 1 < k ≤ n0, n1 ≤ n, the expected difference in the relative frequency of 1 for k/1-streak successors that are 1 and the relative frequency
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