Monty Hall Problem with Asymmetric Host Behavior

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Bayes / Decision Theory.

Your Task
In a modified Monty Hall game, the prize is hidden behind Door 1 with probability $p_1 = 0.5$, Door 2 with probability $p_2 = 0.3$, and Door 3 with probability $p_3 = 0.2$. You choose Door 1. Monty, who knows where the prize is, opens a different door (Door 2 or Door 3) that does NOT contain the prize. If both Doors 2 and 3 are empty, Monty chooses between them uniformly at random. Monty opens Door 3. What is the probability that the prize is behind Door 2? Should you switch?

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.