Question

Events $A$ and $B$ are such that $P\left(A\right)=\frac{1}{2},P\left(B\right)=\frac{7}{12}$ and $P\left(notAornotB\right)=\frac{1}{4}$. State whether $A$ and $B$ are independent ?

• A. True
• B. False

Answer 1

The correct option is B False

It is given that $P\left(A\right)=\frac{1}{2},P\left(B\right)=\frac{7}{12}$ and $P\left(notAornotB\right)=\frac{1}{4}$

$\Rightarrow P(A\prime \cup B\prime )=\frac{1}{4}$

$\Rightarrow P\left((A\cap B)\prime \right)=\frac{1}{4}$ $[A\prime \cup B\prime =(A\cap B)\prime ]$

$\Rightarrow 1-P(A\cap B)=\frac{1}{4}$

$\Rightarrow P(A\cap B)=\frac{3}{4}$ ....(1)

However, $P\left(A\right)\cdot P\left(B\right)=\frac{1}{2}\cdot \frac{7}{12}=\frac{7}{24}$ ....(2)

Here, $\frac{3}{4}\ne \frac{7}{24}$

$\therefore P(A\cap B)\ne P\left(A\right)\cdot P\left(B\right)$
Therefore, $A$ and $B$ are independent events.