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. $P\left(A^{\prime }\right)+P\left(B^{\prime }\right)-1$
• C. $P\left(B\right)-P\left(A^{\prime }\right)$
• D. $P\left(B^{\prime }\right)-P\left(A\right)$

Answer 1

Answer: (C)

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

Let's see the value of 1st option:
$P(\overline{A}\cap \overline{B})-P(A\cap B)=P\left(\overline{A}\right)P\left(\overline{B}\right)-P\left(A\right)P\left(B\right)$
since all the events given are independent.
$=(1-P\left(A\right))(1-P\left(B\right))-P\left(A\right)P\left(B\right)=1-P\left(A\right)-P\left(B\right)=P\left(\overline{A}\right)-P\left(B\right)whichisoptionB=1-P\left(B\right)-P\left(A\right)=P\left(\overline{B}\right)-P\left(A\right)whichisoptionD$
Hence option C is not equal to any of other.