Bertrand's Box Paradox and Conditional Probability

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Conditional Probability.

Your Task
There are three identical boxes, each with two drawers:

• Box 1: contains two gold coins ($GG$).
• Box 2: contains two silver coins ($SS$).
• Box 3: contains one gold coin and one silver coin ($GS$). A box is selected uniformly at random, and one of its drawers is opened, revealing a gold coin. What is the probability that the other drawer in the same box also contains a gold coin?

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.