Question

The rifleman is firing at a distant target and has only $10\%$ chance of hitting it. The least number of rounds he must fire to have more than $50\%$ chance of hitting it at lease once is

• A. $5$
• B. $7$
• C. $9$
• D. $11$

Answer 1

The correct option is B $7$

$p=\frac{1}{10},q=\frac{9}{10}$
$A\to$ he hits the target
$A^{\prime }\to$ Not hitting the target
$P\left(A^{\prime }\right)=q\cdot q\cdot q\cdots q\text{(say}n\text{times)}$
now $P\left(A\right)+P\left(A^{\prime }\right)=1$
$P\left(A\right)=1-P\left(A^{\prime }\right)$
$=1-q^n$
But $P\left(A\right)>\frac{1}{2}\Rightarrow 1-q^n>\frac{1}{2}\Rightarrow \frac{1}{2}>{\left(\frac{9}{10}\right)}^n$
Now $n=6\Rightarrow .5314n=7\Rightarrow .47$
$\therefore$ Requires atleast $7$ shots.