Probability of Truth Given Corroboration
Alice and Bob are witnesses to a crime. When asked questions, Alice tells the truth with probability $p_A$ and lies with probability
- p_A$. Bob tells the truth with probability $p_B$ and lies with probability
- p_B$. Their truthfulness is independent.
The detective asks Alice a yes/no question. She gives an answer, and Bob then states that Alice's answer is truthful.
Given that $p_A = 1/3$ and $p_B = 3/5$, find the probability that Alice actually told the truth.
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