Question

If $A$ and $B$ be two independent events then show that $A,B^{\prime }$ are also independent.

Answer 1

Given $A$ and $B$ are independent events.

Then $P(A\cap B)=P\left(A\right).P\left(B\right)$.......(1).

Now,

$P(A\cap B^{\prime })$

$=P\left(A\right)-P(A\cap B)$ [ From the definition of $A\cap B^{\prime }$]

$=P\left(A\right)-P\left(A\right).P\left(B\right)$ [ Using (1)]

$=P\left(A\right)\{1-P(B\left)\right\}$

$=P\left(A\right).P\left(B^{\prime }\right)$.

So $P(A\cap B^{\prime })=P\left(A\right).P\left(B^{\prime }\right)$.

This gives $A,B^{\prime }$ are independent.