Solution
= Solution
Let $S=\sum_{i=1}^nX_i$ be the number of heads. The <likelihood function> for the <Bernoulli distribution> sample is proportional to $\theta^S(1-\theta)^{n-S}$. A <uniform distribution> prior on $[0,1]$ is the $\operatorname{Beta}(1,1)$ distribution, so <Beta-binomial conjugacy> gives the <posterior distribution>
$$
\boxed{\theta\mid X_1,\ldots,X_n\sim\operatorname{Beta}(S+1,n-S+1)}.
$$