Question

The probability that a missile hits a target successfully is $0.75$. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than $0.95$, is______

Answer 1

Let ' $n$ ' be the number of missiles fired
$\Rightarrow $ probability of at least $3$ successful hits
$=1-{ }^{n} C_{0}\left(\frac{1}{4}\right)^{n}-{ }^{n} C_{1}\left(\frac{3}{4}\right)\left(\frac{1}{4}\right)^{n-1}-{ }^{n} C_{2}\left(\frac{3}{4}\right)^{2}\left(\frac{1}{4}\right)^{n-2}$
$\Rightarrow 1-{ }^{n} C_{0}\left(\frac{1}{4}\right)^{n}-{ }^{n} C_{1}\left(\frac{3}{4}\right)\left(\frac{1}{4}\right)^{n-1}-{ }^{n} C_{2}\left(\frac{3}{4}\right)^{2}\left(\frac{1}{4}\right)^{n-2} \geq 0.95 $
$\Rightarrow 9 n^{2}-3 n+2 \leq \frac{4^{n}}{10} $
$\Rightarrow $ minimum value of '$n$' is '$6' (n \in N)$