#### 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

11

9

7

5

#### Solution

7

Let *p*=chance of hitting a distant target

\[\Rightarrow\] *p* =10% or *p*= 0.1

\[\Rightarrow q = 1 - 0 . 1 = 0 . 9\]

\[\text{ Let n be the least number of rounds } . \]

\[P(\text{ hitting atleast once} ) = P(X \geq 1) \]

\[ \Rightarrow 1 - P(X = 0) \geq 50 \% \]

\[ \Rightarrow 1 - P(X = 0) \geq 0 . 5\]

\[P(X = 0) \leq 0 . 5\]

\[ \Rightarrow (0 . 9 )^n \leq 0 . 5\]

\[\text{ Taking } \text{ log on both the sides, we get} \]

\[ n \text{ log } 0 . 9 \leq \log 0 . 5 \]

\[ \Rightarrow n \leq \frac{\log 0 . 5}{\log 0 . 9}\]

\[ \Rightarrow n \leq 7 . 2 \]

\[\text{ Therefore, 7 is the least number of rounds that he must fire in order } \]

\[ \text{ to have more than 50 % chance of hitting the target at least once } . \]