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Let A and B be the events such that 2P(A) = P(B) = \(\frac{5}{13}\) and P(A|B) = \(\frac{2}{5}\) . Then, find P(A ∪ B) and P(A ∩ B). |
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Answer» We have 2P(A) = P(B) = \(\frac{5}{13}\) and P(A|B) = \(\frac{2}{5}\), therefore, P(A) = \(\frac{P(B)} {2}\) = \(\frac{5}{26}\) . By conditional probability, we know that P(A|B) = \(\frac{P(A∩B)}{P(B)}\). ⇒ P(A ∩ B) = P(A|B)P(B) = \(\frac{2}{5} \times \frac{5}{13} = \frac{2}{13}\). We know that P(A ∩ B)= P(A) + P(B) − P(A ∩ B) = \(\frac{5}{26} + \frac{5}{13} - \frac{2}{13} = \frac{5+10-4}{26} = \frac{11}{26}.\) Hence, P(A ∪ B) = \(\frac{11}{26}\) and P(A ∩ B) = \(\frac{2}{13}\). |
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