Question

A rifleman is firing at a distant target and has only 10% chance of hitting it. The least number of rounds he must fire in order to have more than 50% chance of hitting it at least once is
(a) 11
(b) 9
(c) 7
(d) 5

Answer 1

(c) 7

Let p=chance of hitting a distant target
$\Rightarrow$p =10% or p= 0.1

$$\begin{gathered} \Rightarrow q=1-0.1=0.9 \\ \mathrm{Let}\mathrm{n}\mathrm{be}\mathrm{the}\mathrm{least}\mathrm{number}\mathrm{of}\mathrm{rounds}. \\ P(\mathrm{hitting}\mathrm{atleast}\mathrm{once})=P(X\ge 1) \\ \Rightarrow 1-P(X=0)\ge 50\% \\ \Rightarrow 1-P(X=0)\ge 0.5 \\ P(X=0)\le 0.5 \\ \Rightarrow (0.9{)}^n\le 0.5 \\ \mathrm{Taking}\log \mathrm{on}\mathrm{both}\mathrm{the}\mathrm{sides},\text{we get} \\ \mathit{}n\log 0.9\le \log 0.5 \\ \Rightarrow n\le \frac{\log 0.5}{\log 0.9} \\ \Rightarrow n\le 7.2 \\ \mathrm{Therefore},7\mathrm{is}\mathrm{the}\mathrm{least}\mathrm{number}\mathrm{of}\mathrm{rounds}\mathrm{that}\mathrm{he}\mathrm{must}\mathrm{fire}\mathrm{in}\mathrm{order} \\ \mathrm{to}\mathrm{have}\mathrm{more}\mathrm{than}50\%\mathrm{chance}\mathrm{of}\mathrm{hitting}\text{the target}\mathrm{at}\mathrm{least}\mathrm{once}. \end{gathered}$$