Question

Let $A$ and $B$ two events such that $P\left(A\cap B\right)=\frac{1}{6}, P\left(A\cup B\right)=\frac{31}{45}$ and $P\left(\bar{B}\right)=\frac{7}{10}$ then

• A. $A$ and $B$ are independent
• B. $A$ and $B$ are mutually exclusive
• C. $P\left(\frac{A}{B}\right)< \frac{1}{6}$
• D. $P\left(\frac{B}{A}\right)< \frac{1}{6}$

Answer 1

Answer: (A)

$P\left(\bar{B}\right)=\frac{7}{10} \Rightarrow P\left(B\right)=\frac{3}{10}$
$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right) \Rightarrow P\left(A\right)=\frac{5}{9}$
$\therefore P\left(A\right) \times P\left(B\right)=\frac{5}{9}\times \frac{3}{10}=\frac{1}{6}=P\left(A\cap B\right)$
$\Rightarrow A, B$ are independent