Proposition: Forty shots under the stated binomial conversion

With $n_j=40$, ${\sum }_j{\alpha }_j={\alpha }_{\mathrm{s}}<1$, and the generic error upper bound

no dataset can accept $\epsilon =0.1$.

Proof. The upper binomial limit is nondecreasing in the observed failure count, so its smallest value occurs with zero failures.

The function $a\mapsto 1-a^{1/40}$ is convex on $(0,1]$. Jensen's inequality gives

Now $3^{-1/40}<0.99$, since $0.99^{40}>1-40(0.01)=0.6>1/3$. Hence $\frac{1}{2}{\sum }_j{u}_j>0.015>0.0025$, and $U_{\mathrm{bin}}>0.1$.