Question

For any events $A$ and $B$. Given $P(A \cup B)=0.6, P(A)=P(B), P(B / A)=0.8$. If the value of $P [( A \cup \overline{ B }) \cup(\overline{ A } \cap B )]$ can be expressed as $\frac{ m }{ n }$ where $m$ and $n$ are relatively prime then find the value of $(m+n)$.

Answer 1

$A \cup \overline{ B } =\left\{ a _1+ a _2\right\} \cup\left\{ a _1+ a _4\right\} $
$ =\left\{ a _1+ a _2+ a _4\right\}$
$\overline{ A } \cap B =\left\{ a _3+ a _4\right\} \cap\left\{ a _2+ a _3\right\}$
$ =\left\{ a _3\right\}$
$P[(A \cup \bar{B}) \cup(\bar{A} \cap B)] =\left\{a_1+a_2+a_4\right\} \cup\left\{a_3\right\} $
$ =\left\{a_1+a_2+a_3+a_4\right\}=1=\frac{1}{1} \quad \Rightarrow \quad m+n=2$