Question

There are three events $A,B,C$ one which must and only one can happen; the odds are $8$ to $3$ against $A$, $5$ to $2$ against $B$. The odds against $C$ is $43:17k$. Find the value of $k$ ?

Answer 1

According to the question
$\frac{P\left(A^{\prime }\right)}{P\left(A\right)}=\frac{8}{3}\Rightarrow P\left(A\right)=\frac{3}{11}$ and $P\left(A^{\prime }\right)=\frac{8}{11}$
$\frac{P\left(B^{\prime }\right)}{P\left(B\right)}=\frac{5}{2}\Rightarrow P\left(B\right)=\frac{2}{7}$ and $P\left(B^{\prime }\right)=\frac{5}{7}$
Now out of $A,B$ and $C$, one and only one can happen.
$P\left(A\right)+P\left(B\right)+P\left(C\right)=1\Rightarrow P\left(C\right)=\frac{34}{77}$
So odd against $C=\frac{P\left(C\right)}{P\left(C^{\prime }\right)}=\frac{43}{34}$
$\therefore k=2$