Question

If $A$ and $B$ are two independent events, then which of the following is not equal to any of the remaining?

• A. ${P(A}^{\prime }\cap B^{\prime })-P(A\cap B)$
• B. ${P(A}^{\prime })+{P(B}^{\prime })-1$
• C. $PB-{PA}^{\prime }$
• D. ${P(B}^{\prime })-P(A)$

Answer 1

The correct option is C $PB-{PA}^{\prime }$

A and B are two independent events
$\therefore P(A\bigcap B)=P\left(A\right).P\left(B\right)$ ..... $\left(i\right)$
As $A$ and $B$ are independent

$⟹A^{{}^{\prime }}\&B^{{}^{\prime }}$ are also independent

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

Consider, $P(A^{{}^{\prime }}\bigcap B^{{}^{\prime }})-P(A\bigcap B)$
$=P\left(A^{{}^{\prime }}\right)P\left(B^{{}^{\prime }}\right)-P\left(A\right)P\left(B\right)(\because P(A)+P(A^{{}^{\prime }})=1)$
$=[1-P(A\left)\right][1-P(B\left)\right]-P\left(A\right)P\left(B\right)$
$=1-P\left(A\right)-P\left(B\right)+P\left(A\right)P\left(B\right)-P\left(A\right)P\left(B\right)$
$=1-P\left(A\right)-P\left(B\right)$
$=[1-P(B\left)\right]-P\left(A\right)$
$=P\left(B^{{}^{\prime }}\right)-P\left(A\right)$

$\therefore P(A^{{}^{\prime }}\bigcap B^{{}^{\prime }})-P(A\bigcap B)=P\left(B^{\prime }\right)-P\left(A\right)$ .... $\left(i\right)$

Also,
$P\left(B^{\prime }\right)-P\left(A\right)=P\left(B^{{}^{\prime }}\right)-P\left(A\right)+1-1$
$=P\left(B^{{}^{\prime }}\right)+[1-P(A\left)\right]-1$
$=P\left(B^{{}^{\prime }}\right)+P\left(A^{{}^{\prime }}\right)-1$ $\to \left(B\right)$

$\therefore$ Option $\left(C\right)\to P\left(B\right)-P\left(A^{{}^{\prime }}\right)$ is not equal to any of the options