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

Answer: 2

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