Question

A hunters 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 $r=3a,4a,.....$ If he misses at a distance na, the animal escapes. Find the odds against the hunter

• A. $n-1:n+1$
• B. $n+1:n-1$
• C. $n:n-1$
• D. $n+1:n$

Answer 1

Answer: (B)

$n+1:n-1$

$E_r$ : hunters shoots the animal at $r$ distance

$P\left(E\right)=P\left(escapes\right)=1-P({\overline{E}}_{2a}{\overline{E}}_{3a}.....{\overline{E}}_{na})$

$=1-(1-\frac{a^2}{4a^2})(1-\frac{a^2}{9a^2})(1-\frac{n{a}^2}{n^2{a}^2})$

$=1-(1-\frac{1}{2^2})(1-\frac{1}{3^2}).....(1-\frac{1}{n^2})$

$=1-\left(\frac{1.3}{2.2}\right)(\frac{2}{3}.\frac{4}{3})(\frac{3}{4}.\frac{5}{4}).....\left(\frac{n+1}{n}\frac{n+1}{n}\right)$

$=1-\frac{n+1}{2n}=\frac{n-1}{2n}$

$\therefore$ odds against the hunter $=\frac{P\left(\overline{E}\right)}{P\left(E\right)}=\frac{n+1}{n-1}$