Question

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

• A. 7
• B. 8
• C. 9
• D. 6

Answer 1

The correct option is A 7

The probability of hitting in one shot = $\frac{10}{100}=\frac{1}{10}$
If he fires n shots, the probability of hitting at least once
$=1-{(1-\frac{1}{10})}^n=1-{\left(\frac{9}{10}\right)}^n=\frac{1}{2}$ (from the question)
$\therefore {\left(\frac{9}{10}\right)}^n=\frac{1}{2},\therefore n\{2{\log }_{10}3-1\}=-{\log }_{10}2$
$\therefore n=\{\begin{array}{c}\frac{{\log }_{10}2}{1-12{\log }_{10}3}\end{array}=\frac{0.3010}{1-2\times 0.4771}=6.5$ (nearly)
$\therefore$ For 6 shots, the probability is about $53\%$ while for 7 shots it is nearly $48\%.$