Generalized Birthday Problem and Poisson Approximation

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Combinatorics / Poisson Approximation.

Your Task
In a room of $n$ people, assume each person's birthday is independently and uniformly distributed across $D = 365$ days. Derive the exact probability $P(n, D)$ that at least two people share a birthday. Show that for $n = o(D^{2/3})$, $P(n, D) \approx 1 - \exp\left(-\frac{n(n-1)}{2D}\right)$ and find the smallest $n$ such that $P(n, 365) \ge 0.5$.

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.