Question

The chance of one event happening is the square of the chance of a $2^{nd}$ event, but odds against the first are the cubes of the odds against the 2nd. Find the chances of first event. (Assume that both events are neither sure nor impossible)

• A. $\frac{1}{9}$
• B. $\frac{1}{3}$
• C. $\frac{2}{3}$
• D. $\frac{2}{9}$

Answer 1

The correct option is A $\frac{1}{9}$

Let the first event $A_1$ and the second event $A_2$
Let $P\left(A_1\right)=\frac{p^2}{q^2}$ $\therefore P\left(A_2\right)=\frac{p}{q}$
odds against second event $=\frac{q-p}{p}$
odds against first event $=\frac{q^2-p^2}{p^2}$
$\Rightarrow {\left(\frac{q-p}{p}\right)}^3=\frac{q^2-p^2}{p^2}$
$\Rightarrow (q-p{)}^2=(q+p)p$
$\Rightarrow q^2=3pq\Rightarrow \frac{p}{q}=\frac{1}{3}$
$\therefore P\left(A_1\right)=\frac{1}{9};P\left(A_2\right)=\frac{1}{3}$