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Solve the following.
If A and B are two independent events such that P(A̅ ∪̅ B̅) = 1/3 and P(A̅) = 2/3, calculate P(B).
If $A$ and $B$ are two independent events such that $P(\overline{A \cup B}) = \frac{1}{3}$ and $P(\bar{A}) = \frac{2}{3}$, calculate $P(B)$.
Answer
Answer
$P(A \cup B) = 1 - \frac{1}{3} = \frac{2}{3}$, $P(A) = 1 - \frac{2}{3} = \frac{1}{3}$.
For independent events, $P(A \cup B) = P(A) + P(B) - P(A)P(B)$.
$\frac{2}{3} = \frac{1}{3} + P(B) - \frac{1}{3}P(B)$
$\frac{2}{3}P(B) = \frac{1}{3}$
Hence $P(B) = \frac{1}{2}$.
Final answer: $\frac{1}{2}$