Q.If $P(B) = \dfrac{3}{5}$, $P(A \mid B) = \dfrac{1}{2}$ and $P(A \cup B) = \dfrac{4}{5}$, then $P((A \cup B)') + P(A' \cup B)$ equals
(A) $\dfrac{1}{5}$
(B) $\dfrac{4}{5}$
(C) $\dfrac{1}{2}$
(D) $1$

Given $P(B)=\tfrac35,\ P(A\mid B)=\tfrac12,\ P(A\cup B)=\tfrac45.$

$P(A\cap B)=P(A\mid B)\,P(B)=\tfrac12\cdot\tfrac35=\tfrac{3}{10},$ and $P(A)=\tfrac45-\tfrac35+\tfrac{3}{10}=\tfrac12.$

$P((A\cup B)')=1-P(A\cup B)=1-\tfrac45=\tfrac15.$ …

Setup

We are given $P(B)=\tfrac35,\ P(A\mid B)=\tfrac12,\ P(A\cup B)=\tfrac45$. First recover $P(A\cap B)$ and $P(A)$:

First term

By the complement rule, $P((A\cup B)')=1-P(A\cup B)=1-\tfrac45=\tfrac15.$

Second term

Use De Morgan's law: $(A'\cup B)'=A\cap B'$, so $P(A'\cup B)=1-P(A\cap B').$

Since $A$ splits into the parts inside and outside $B$, $P(A\cap B')=P(A)-P(A\cap B)=\tfrac12-\tfrac{3}{10}=\tfrac15.$ …

Method: Complement rule and De Morgan for compound expressions

Use this for expressions built from unions, complements and conditionals that must each be simplified before adding.

Steps

Step 1: Simplify each complemented compound with De Morgan / complement rule.

So $P(A'\cup B)=1-P(A\cap B')$.

Step 2: Reduce the leftover joint terms to known quantities. …

Common Mistakes

Mistake 1: Writing $P(A'\cup B)=1-P(A\cup B)$.

Why it's wrong: the complement of $A\cup B$ is $A'\cap B'$, not $A'\cup B$. Correct approach: use De Morgan — $(A'\cup B)'=A\cap B'$, so $P(A'\cup B)=1-P(A\cap B')$.

Mistake 2: Leaving $P(A\cap B')$ unreduced. …