Let \(X\) be the number of patients in a clinical trial with a successful outcome. Let \(P\) be the probability of success for an individual patient. We assume before the trial begins that \(P\) is unifom on \([0,1]\). Compute

1. \(f(P \mid X)\)
2. \( {\mathbf E}( P \mid X)\)
3. \( {\mathbf Var}( P \mid X)\)