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Solve the following.
Given that the events A and B are such that P(A) = 1/2, P(A ∪ B) = 3/5 and P(B) = k. Find k if: A…
Given that the events $A$ and $B$ are such that $P(A) = \frac{1}{2}$, $P(A \cup B) = \frac{3}{5}$ and $P(B) = k$. Find $k$ if:
(a)[1]
$A$ and $B$ are mutually exclusive.
(b)[1]
$A$ and $B$ are independent.
Answer
Answer (a)
Mutually exclusive: $P(A \cap B) = 0$, so $P(A \cup B) = P(A) + P(B)$.
$\frac{3}{5} = \frac{1}{2} + k$
Hence $k = \frac{1}{10}$.
Final answer: $\frac{1}{10}$
Answer (b)
Independent: $P(A \cap B) = P(A)P(B) = \frac{k}{2}$.
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
$\frac{3}{5} = \frac{1}{2} + k - \frac{k}{2}$, so $\frac{k}{2} = \frac{1}{10}$
Hence $k = \frac{1}{5}$.
Final answer: $\frac{1}{5}$