= Solution
Conditional on the success probability, the count has <binomial distribution>. Integrating its <likelihood> under the uniform <prior distribution>, or equivalently a <beta distribution> with parameters $(1,1)$, gives
$$
p(y\mid H_1)=\int_0^1\binom ny\theta^y(1-\theta)^{n-y}\,d\theta
=\binom ny\frac{\Gamma(y+1)\Gamma(n-y+1)}{\Gamma(n+2)}
=\boxed{\frac1{n+1}},\qquad y=0,\ldots,n.
$$
The <prior predictive distribution> is therefore uniform over the possible counts. Excluding the single point $\theta=1/2$ does not alter this integral because a continuous <prior distribution> gives that point zero probability. This calculation is the marginalization underlying <Beta-binomial conjugacy>.
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