Question

A man firing at a distant target has $10\%$ chance of hitting the target in one shot. The number of times he must fire at the target to have about $50\%$ chance of hitting the target is

• A. $11$
• B. $9$
• C. $7$
• D. $5$

Answer 1

Answer: (C)

$7$

The probability of hitting in one shot $=\frac{10}{100}=\frac{1}{10}$
$\therefore$ The probability of not hitting in one shot $=1-\frac{1}{10}$
If he fgires $n$ shots, the probability of hitting at least once
$=1-{(1-\frac{1}{10})}^n=1-{\left(\frac{9}{10}\right)}^n=\frac{1}{2}$ (as given)
$\Rightarrow {\left(\frac{9}{10}\right)}^n=\frac{1}{2}\Rightarrow n(\log 9-\log 10=\log 1-\log 2)$
$\Rightarrow n(2\log 3-1)=-\log 2\Rightarrow n=\frac{\log 2}{1-2\log 3}$
$=\frac{0.3010}{1-2\left(0.4771\right)}=\frac{0.3020}{1-0.9542}=\frac{0.3010}{0.0452}=6.6$ (nearly)
$\therefore$ for $6$ shots, the probability is about $53$% for $7$ shots it is nearly $48$%.
Hence $n=7$.