Question

The odds against an event is 4:5 and the odds in favour of another event is 3:7. If both the events are independent, then the probability that at least one of the event will happen is

• A. $\frac{31}{45}$
• B. $\frac{77}{90}$
• C. $\frac{1}{6}$
• D. $\frac{5}{6}$

Answer 1

The correct option is A $\frac{31}{45}$

$P\left(\overline{A}\right):P\left(A\right)=4:5\Rightarrow P\left(A\right)=\frac{5}{9}$
$P\left(B\right):P\left(\overline{B}\right)=3:7\Rightarrow P\left(B\right)=\frac{3}{10}$
$=P(A\cup B)\Rightarrow P\left(A\right)+P\left(B\right)-P(A\cap B)$
$=P\left(A\right)+P\left(B\right)-P\left(A\right)P\left(B\right)$
$=\frac{5}{9}+\frac{3}{10}-\frac{5}{9}.\frac{3}{10}=\frac{31}{45}$