Compute P(Bag B | red) via Bayes

There are three bags containing red (r) and green (g) balls:

• Bag A: 4 r, 6 g
• Bag B: 6 r, 4 g
• Bag C: 3 r, 7 g Procedure: choose one bag uniformly at random, then draw one ball from that bag. Given that the drawn ball is red, compute the posterior probability that it came from Bag B, i.e., $P(B \mid red)$. Show all steps using Bayes' rule. Follow-up: if instead the bag is chosen with probabilities $P(A)=0.2$, $P(B)=0.5$, $P(C)=0.3$, recompute $P(B \mid red)$ and explain the intuition for how the prior changes the result. Overview: This question tests understanding of Bayes' rule and conditional probability by requiring the posterior probability that a particular bag was chosen given a red draw.