Mean exceedance for Bernoulli sums with one exceptional weight

Let B0,...,Bn be independent Bernoulli variables with common parameter p. A candidate proof is given that Pr[aB0 + b(B1 + ... + Bn) >= p] >= p for a >= 0, b > 0, a + nb = 1 and 0 <= p <= 1/3, for every finite n >= 1. The argument uses a capacity reduction, a binomial endpoint inequality, an exact splitting identity and reflected binomial mass comparisons. The biases for which the bound holds throughout this weight class are [0,1/3] union {1/2,1}. The general weighted conjecture and a uniform positive surplus are not claimed. A related-literature coverage limitation and residual priority uncertainty are documented in the article and supporting record.