In a bombing attack, there is $50 \%$ chance that a bomb will hit the target. At least two independe

Question

In a bombing attack, there is $50 \%$ chance that a bomb will hit the target. At least two independent hits are required to destroy the target completely. Then the minimum number of bombs, that must be dropped to ensure that there is at least $99 \%$ chance of completely destroying the target, is

Professor PC Thomas Classes · Answer

(11) : Probability of hitting the target, $p=\frac{1}{2}$\ Probability of non-hitting the target, $q=1-\frac{1}{2}=\frac{1}{2}$\ Let $n$ bombs are required to destroy the target. According to question, we have\ $1-\left({ }^{n} C_{0} p^{0} q^{n}+{ }^{n} C_{1} p^{1} q^{n-1}ight) \geq \frac{99}{100}$\ $\Rightarrow 1-\frac{1}{2^{n}}-\frac{n}{2^{n}} \geq \frac{99}{100} \Rightarrow \frac{1}{100} \geq \frac{n+1}{2^{n}}$\ $\Rightarrow \quad 2^{n} \geq(n+1) 100$\ At $n=10,2^{10} \geq 1100$; not possible\ At $n=11,2^{11} \geq 1200$ which is true.\