Laplace's Rule of Succession and Beta-Binomial Inference

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Bayesian Inference.

Your Task
A coin with unknown probability of heads $\theta \in [0, 1]$ has a flat prior $\theta \sim \text{Uniform}(0, 1) = \text{Beta}(1, 1)$. The coin is tossed $n$ times and lands heads $k$ times.
(1) Derive the posterior distribution $\theta \mid k, n$.
(2) Derive Laplace's Rule of Succession: the predictive probability that the $(n+1)$-th toss will also land heads is $\frac{k+1}{n+2}$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.