Question

A fireman is firing at a distant target and has only $10\%$ chance of hitting it. The number of rounds, he must fire in order to have $50\%$ chance of hitting it at least once is

• A. $5$
• B. $7$
• C. $9$
• D. $11$

Answer 1

The correct option is B $7$

For the first time the chance of hitting is $\frac{1}{10}$.

For the second time the chance of hitting is $\frac{9}{10}\times \frac{1}{10}$.

Let us assume that it takes $x$ rounds to get $50$% chance of hitting

We have $\frac{1}{10}+\frac{9}{10}\times \frac{1}{10}+.........+(\frac{9}{10}{)}^{(x-1)}\times \frac{1}{10}\ge \frac{1}{2}$

$\Rightarrow \frac{1}{10}\left(c\frac{9}{10}\right(\frac{1-(\frac{9}{10}{)}^{(x-1)}}{1-\frac{9}{10}}\left)\right)\ge \frac{1}{2}$

$\Rightarrow \frac{9}{10}-(\frac{9}{10}{)}^x\ge \frac{1}{2}$

Therefore for $x=7$ , there will be $50$% chance of hitting it atleast once.

Therefore the correct option is $B$.