Question

Events A and C are independent. If the probability relating A, B and C are $P\left(A\right)=1/5,P\left(B\right)=1/6;$ $P(A\cap C)=1/20.P(B\cup C)=3/8$. Then

• A. events B and C are independent
• B. events B and C are mutually exclusive
• C. events B and C are neither independent nor mutually exclusive
• D. events B and C are equiprobable

Answer 1

The correct option is A events B and C are independent

Given $P\left(A\right)=\frac{1}{5},P\left(B\right)=\frac{1}{6},P(A\cap C)=\frac{1}{20}$and $P(B\cup C)=\frac{3}{8}$
since A and C are independent
$\Rightarrow P(A\cap C)=P\left(A\right)P\left(C\right)=\frac{1}{20}\Rightarrow P\left(C\right)=\frac{1}{20}\times 5=\frac{1}{4}\Rightarrow P\left(C\right)=\frac{1}{4}AlsoP(B\cup C)=\frac{3}{8}\Rightarrow P\left(B\right)+P\left(C\right)-P(B\cap C)=\frac{3}{8}\Rightarrow P(B\cap C)=\frac{1}{6}+\frac{1}{4}-\frac{3}{8}=\frac{1}{24}\Rightarrow P(B\cap C)=P\left(B\right)P\left(C\right)=\frac{1}{24}$
Which means events $B$ and $C$ are independent