Question

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

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

Answer 1

Answer: (A)

$A$ and $B$ are independent

We have, $P(A\cap B)=\frac{1}{6},P(A\cup B)=\frac{31}{45}$ and $P\left(\overline{B}\right)=\frac{7}{10}$

$\therefore P\left(B\right)=1-\frac{7}{10}=\frac{3}{10}$

Now using $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$\Rightarrow \frac{31}{45}=P\left(A\right)+\frac{3}{10}-\frac{1}{6}=P\left(A\right)+\frac{2}{15}$

$\Rightarrow P\left(A\right)=\frac{31}{45}-\frac{2}{15}=\frac{25}{45}=\frac{5}{9}$

Check, $P\left(A\right)P\left(B\right)=\frac{5}{9}\cdot \frac{3}{10}=\frac{1}{6}=P(A\cap B)$

Hence both the events $A$ and $B$ are independent to each other