 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 $\mathrm{Beta}(a,a)$ with $a>1$ is centered at $1/2$, withChoosing $a=50$ 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.