Mathematics - Probability Question with Solution | TestHub

Let the number of missiles be $n$,

$\mathrm{P}$ (of destroying the target) $\ge 0.95$

${}^{n}C_3·{\left(\frac{3}{4}\right)}^3·{\left(\frac{1}{4}\right)}^{n-3}+{}^{n}C_4{\left(\frac{3}{4}\right)}^4{\left(\frac{1}{4}\right)}^{n-4}+\dots ..+{}^{n}C_n{\left(\frac{3}{4}\right)}^n\ge 0.95$

$1-\left\{{}^{n}C_0{\left(\frac{3}{4}\right)}^0·{\left(\frac{1}{4}\right)}^n+{}^{n}C_1{\left(\frac{3}{4}\right)}^1·{\left(\frac{1}{4}\right)}^{n-1}+{}^{n}C_2{\left(\frac{3}{4}\right)}^2{\left(\frac{1}{4}\right)}^{n-2}\right\}\ge 0.95$

$1-\frac{95}{100}\ge \frac{1}{4^n}+\frac{3n}{4^n}+\frac{n\left(n-1\right)}{2}·\frac{9}{4^n}$

$\frac{4^n}{20}\ge \frac{2+6n+9n^2-9n}{2}$

$\frac{2^{2n-1}}{5}\ge 9n^2-3n+2$

$2^{2n-1}\ge 5\left(9n^2-3n+2\right)$

$n=3\Rightarrow 32\ge 5\times 74$   (not true)

$n=4\Rightarrow 128\ge 5\times 134$   (not true)

$n=5\Rightarrow 512\ge 5\times 212$    (not true)

$n=6\Rightarrow 2048\ge 5\times 308$    (true)

So, $n=6$.