Aggregate Success Rate in Stopped Bernoulli Trials: Expectation, Monotonicity, and Convergence

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 → ∞.