Let A and B be the events such that 2P(A) = P(B) = 5 /13 and P(A|B) = 2 /5 . Then, find P(A ∪ B) and P(A ∩ B).

Question

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).

Answer 1

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) = \(\frac{2}{13}\).