Mathematics - Probability Question with Solution | TestHub

Given $n=5,p=\frac{1}{4}$ and $q=1-p=\frac{3}{4}$

Required probability $=$ when no machine has fault $+$ when one machine has fault $+$ when two machines have fault.

$={}_{}{}^{5}C_0{\left(\frac{3}{4}\right)}^5+{}_{}{}^5\mathrm{C}_1\left(\frac{1}{4}\right){\left(\frac{3}{4}\right)}^4+{}_{}{}^5\mathrm{C}_2{\left(\frac{1}{4}\right)}^2{\left(\frac{3}{4}\right)}^3$

$=\frac{243}{1024}+\frac{405}{1024}+\frac{270}{1024}=\frac{918}{1024}=\frac{459}{512}=\frac{27\times 17}{64\times 8}$

$\Rightarrow {\left(\frac{3}{4}\right)}^3\times k={\left(\frac{3}{4}\right)}^3\times \frac{17}{8}$

$\therefore k=\frac{17}{8}$