Three boxes are equally likely to be selected: - Box A contains two red balls. - Box B contains one red and one blue ball. - Box C contains two blue balls. Draw one ball uniformly from the selected box without replacement and observe red. 1. What is the posterior probability that Box A was selected? 2. What is the probability the second draw is also red? 3. If the second draw is observed to be red, what is the posterior probability that both balls came from Box A? ### Constraints & Assumptions - The box is selected once and is not changed between draws. - Both balls in a box are equally likely to be drawn first. - Draws are without replacement. ```hint Condition on the evidence Weight each prior box probability by the likelihood of observing the first red ball. ``` ### What a Strong Answer Covers - Bayes' rule with explicit priors and likelihoods. - The distinction between conditioning after one red and after two reds. - A simple event-tree or likelihood calculation that makes impossible paths visible. ### Follow-up Questions - How do the answers change if the boxes are selected with unequal probabilities? - What if the first ball is replaced before the second draw? - Which result is the classic source of the common intuition error?