= Solution
An alternative that allows small departures from chance, while assigning little probability to near-perfect prediction or near-perfect failure, is more plausible than a uniform <prior distribution>. For example, a symmetric <beta distribution> $\operatorname{Beta}(a,a)$ with $a>1$ is centered at $1/2$, with
$$
\mathbb E\theta=\frac12,\qquad \operatorname{Var}(\theta)=\frac1{4(2a+1)}.
$$
Choosing $a=50$ gives a prior standard deviation of about $0.05$ and concentrates on modest deviations. Its concentration should represent substantive beliefs specified before examining this experiment, rather than being tuned to the observed count. A continuous <prior distribution> centered at $1/2$ still places no point mass at $H_0$; the separate model <prior probability> supplies that mass.
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