Question

A hunter's chance of shooting an animal at a distance $r$ is $\frac{a^2}{r^2}(r>a).$ He fires when $r=2a$ if he misses he reloads fires when $r=3a,4a,\cdots$. If he misses at a distance $na$, the animal escapes, then the odds against the hunter is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

• A. $\frac{n+1}{n-1}$
• B. $\frac{n-1}{n+1}$
• C. $\frac{n-1}{n}$
• D. $\frac{n+1}{n}$

Answer 1

The correct option is A $\frac{n+1}{n-1}$

$P\left(\text{Animal is not shot}\right)=\prod _{r=2}^n(1-\frac{a^2}{(ra{)}^2})=\prod _{r=2}^n\left(\frac{r^2-1}{r^2}\right)=\prod _{r=2}^n\left(\frac{r+1}{r}\right)\times \prod _{r=2}^n\left(\frac{r-1}{r}\right)=\frac{n+1}{2}\times \frac{1}{n}=\frac{n+1}{2n}$

Therefore, the odds against him
$=\frac{\frac{n+1}{2n}}{1-\frac{n+1}{2n}}=\frac{n+1}{n-1}$