Two Boxes of Balls

Write down $P(\text{red}|A)$ and $P(\text{red}|B)$ before combining them.

Apply Bayes' theorem with equal priors of $1/2$ for each box.

1. 1. Likelihoods

   $P(\text{red}|A)=2/3$ and $P(\text{red}|B)=1/3$, with priors $P(A)=P(B)=1/2$.

2. 2. Bayes' theorem

   $P(A|\text{red})=\frac{(2/3)(1/2)}{(2/3)(1/2)+(1/3)(1/2)}=\frac{1/3}{1/2}=\frac{2}{3}$.

Answer: 2/3