Mathematics - Probability Question with Solution | TestHub

$P(A\cap B)=0.1,P(A\mid B)$ and $P(B\mid A)$ are the roots of the equation $12x^2-7x+1=0$
$\begin{array}{rl}& \Rightarrow P(A\mid B)P(B\mid A)=\frac{1}{12}\\ & \Rightarrow \frac{P(A\cap B)}{P(B)}\times \frac{P(A\cap B)}{P(A)}=\frac{1}{12}\\ & \Rightarrow P(A)P(B)=12(0.1{)}^2\\ & =0.12\end{array}$

Also, $P(A\mid B)+P(B\mid A)=\frac{7}{12}$
$\begin{array}{rl}\Rightarrow & P(A\cap B)\left(\frac{1}{P(B)}+\frac{1}{P(A)}\right)=\frac{7}{12}\\ \Rightarrow & P(A)+P(B)=\frac{7}{12}\times \frac{0.12}{0.1}\\ \Rightarrow & P(A)+P(B)=0.7\\ & \frac{P(\bar{A}\cup \bar{B})}{P(\bar{A}\cap \bar{B})}=\frac{P(\overline{A\cap B})}{P(\overline{AUB})}\\ & =\frac{1-P(A\cap B)}{1-P(A\cup B)}\\ & =\frac{1-0.1}{1-(0.7-0.1)}=\frac{0.9}{0.4}=\frac{9}{4}\end{array}$