Compute conditional occupancy across two rooms

There are two rooms, and the prior distribution over their occupancy is: with probability $\frac{1}{3}$ both rooms are occupied, with probability $\frac{1}{3}$ exactly one room is occupied, and with probability $\frac{1}{3}$ both rooms are empty. You choose one of the rooms uniformly at random and inspect it. (a) What is the probability that the room you inspect is occupied? (b) If the room you inspect turns out to be occupied, what is the probability that the other room is occupied as well? Show the Bayes calculation.

Overview: This question assesses a data scientist's grasp of conditional probability and Bayesian updating, probing probabilistic reasoning and inference about joint occupancy states.