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Question

# A rifleman is firing at a distant target and has only $10%$ chance of hitting it. The minimum number of round he must fire in order to have $50%$chance of hitting it at least once i

A

$7$

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B

$8$

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C

$9$

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D

$6$

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Solution

## The correct option is A $7$Explanation for the correct option:Step 1 : Find probability of hitting target in $1$ shot and that of not hitting the target The probability of hitting the target in $1$ shot $=\frac{10}{100}$ $=\frac{1}{10}$ The probability of not hitting the target $=1-\left(\frac{1}{10}\right)$ $=\frac{9}{10}$Step 2. : Find the probability of hitting once in $\text{'}n\text{'}$ shotsThe probability of hitting once in $\text{'}n\text{'}$ shots $=1-{\left(\frac{9}{10}\right)}^{n}$According to question,$1-{\left(\frac{9}{10}\right)}^{n}=\frac{1}{2}$$⇒$ ${\left(\frac{9}{10}\right)}^{n}=\frac{1}{2}$Step 3. Taking log on both the sides $n\mathrm{log}\frac{9}{10}=\mathrm{log}\frac{1}{2}$$⇒$ $n=\frac{\mathrm{log}0.5}{\mathrm{log}0.9}$$⇒$ $n=6.58~\mathbf{7}$Hence, Option ‘A’ is Correct.

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