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. $10$
• B. $9$
• C. $7$
• D. $8$

Answer 1

The correct option is C

$7$

Explanation for the correct option:

Option (C): $7$

Here, the probability of hitting a target in one shot = $\frac{10}{100}$

= $\frac{1}{10}$

The probability of not hitting a target in one shot = $1-\frac{1}{10}$

= $\frac{9}{10}$

Let the number of shots fired by the man be x,

Given, $1-{\left(\frac{9}{10}\right)}^x=50\%$

$$\begin{gathered} \Rightarrow 1-{\left(\frac{9}{10}\right)}^x=\frac{1}{2} \\ \Rightarrow 1-\frac{1}{2}={\left(\frac{9}{10}\right)}^x \\ \Rightarrow \frac{1}{2}={\left(\frac{9}{10}\right)}^x \\ \Rightarrow \log \left(\frac{1}{2}\right)=x\log \left(\frac{9}{10}\right) \\ \Rightarrow \log \left(1\right)-\log \left(2\right)=x(\log \left(9\right)-\log \left(10\right)) \\ \Rightarrow 0-0.3010=x(0.9542-1) \\ \Rightarrow -0.3010=x(-0.0458) \\ \Rightarrow x=\frac{0.3010}{0.0458} \\ \Rightarrow x=6.572 \end{gathered}$$

$\Rightarrow x=7$

Therefore, the man has to fire $7$ times to have a $50\%$ chance of hitting the target..

is not $10$.

Hence, Option (C) is the correct answer.