Question

The odds against a certain event are $5$ to $2$, and the odds in favour of another event independent of the former are $6$ to $5$: find the chance that one at least of the events will happen.

Answer 1

let the two events be $E_1$ and $E_2$

Odd against favour of one are $5:2$

$\Rightarrow$ odds in favour are $2:5$

$\Rightarrow$ Probability of happenig of one $=P\left(E_1\right)=\frac{2}{7}$

$\Rightarrow$ probabiility of not happening one $=P\left(\stackrel{\_}{E_1}\right)=1-\frac{2}{7}=\frac{5}{7}$

odds in favour of two are $6:5$

$\Rightarrow P\left(E_2\right)=\frac{6}{11}$

$\Rightarrow P\left(\stackrel{\_}{E_2}\right)=1-\frac{6}{11}=\frac{5}{11}$

Desired probabililty $=P\left(E\right)$=$1-$none of then happens

$\Rightarrow P\left(E\right)=1-\frac{5}{7}\times \frac{5}{11}=\frac{52}{77}$