Question

The chance of an event happening is the square of the chance of a second event but the odds against the first are the cube of the odds against the second. The chance of happening of each event are
(Where, $p_1,p_2$ be the chances of happening of the first and second events, respectively)

• A. $p_1=\frac{1}{9}$
• B. $p_1=\frac{1}{16}$
• C. $p_2=\frac{1}{3}$
• D. $p_2=\frac{1}{4}$

Answer 1

Answer: (A), (C)

$p_2=\frac{1}{3}$

Given $p_1,p_2$ be the chances of happening of the first and second events, respectively. Then according to the given conditions, we have
$p_1=p_2^2$
and $\frac{1-p_1}{p_1}={\left(\frac{1-p_2}{p_2}\right)}^3$
$\Rightarrow \frac{1-p_2^2}{p_2^2}={\left(\frac{1-p_2}{p_2}\right)}^3\Rightarrow p_2(1+p_2)=(1-p_2{)}^2$
$\Rightarrow 3p_2=1\Rightarrow p_2=\frac{1}{3}$
Therefore, $p_1=\frac{1}{9}$