Expected Cards Dealt Until First Ace

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Symmetry / Expectation.

Your Task
A standard, well-shuffled 52-card deck is dealt one card at a time. What is the expected number of cards dealt until the first Ace appears (including the Ace itself)?

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.