Mathematical analysis demonstrates aggregate success rates exceed underlying probabilities in restricted geometric experiments, indicating an upward bias that converges as repetitions increase.
We consider m independent sequences of Bernoulli trials with success probability p, each terminated at its first success or after n trials, whichever occurs first. Let Xm,n,p denote the proportion of successes among all trials performed across the m components. We derive a unified integral representation for E(Xm,n,p) that covers both finite cutoffs and the unrestricted case n = ∞. For every m ≥ 1, n ∈ {2, 3, ...} ∪ {∞}, and 0 < p < 1, we prove that E(Xm,n,p) > p. For each such n and p, the sequence m ↦ E(Xm,n,p) is strictly decreasing and strictly convex in the discrete sense. We also establish almost-sure convergence of Xm,n,p to p and convergence of E(Xm,n,p) to p as m → ∞. Finally, for fixed m ≥ 1 and n ∈ {1, 2, ...} ∪ {∞}, E(Xm,n,p) is strictly increasing in p. These results describe the expectation and structural behavior of the aggregate success rate in repeated restricted geometric experiments.
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Yun Soo Kim (2026) studied this question.