Find P(C) - Mutually Exclusive Events and Conditional Probability

Question

Events $A, B, C$ are such that $B$ and $C$ are mutually exclusive and $P(A) = 2/3, P(A \cup B) = 5/6$ and $P(B \cup C) = 4/5$. If $P(B|A) = 1/2$ and $P(C|A) = 3/10$, calculate $P(C)$.

Answer 1

You correctly guessed that we need $P(B)$ first. Since $P(A)$ is known, we should look for relations between $A,B$.

They are $P(A\cup B)=5/6,P(B|A)=1/2$.

By definition$P(B|A)=P(A\cap B)/P(A)$ which gives $P(A\cap B)=\frac12\times P(A)=1/3$.

We know $P(A\cup B)=P(A)+P(B)-P(A\cap B)=5/6$ using which find $P(B)$.

Finally $P(B\cup C)=P(B)+P(C)$ since $P(B\cap C)=0,$ using which find $P(C)$.