= Solution
\b[Direct independent sampling is preferable.] The <posterior distribution> is the standard <beta distribution> $\operatorname{Beta}(9,38)$, for which exact <posterior moments> and standard random-number generators are available. For example, draw independent $U\sim\operatorname{Gamma}(9,1)$ and $V\sim\operatorname{Gamma}(38,1)$ with the same rate, and return $U/(U+V)$. This has the required <beta distribution> and avoids <Markov chain Monte Carlo> convergence problems. A correct, converged MCMC method could sample this target, but it is unnecessarily complicated here.
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