Mathematics - Probability Question with Solution | TestHub

Let $k_1=4{\lambda }_1+r_1,r_1\in \{0,1,2,3\}$
$k_2=4{\lambda }_2+r_2$
$\begin{array}{rl}& (i{)}^{k_1}+(i{)}^{k_2}=(i{)}^{r_1}+(i{)}^{r_2}\\ & (i{)}^{r_1}\in \{1,i,-1,-i\}\end{array}$
$\text{ Zero }\Rightarrow 1,(-1)\text{ pair }\Rightarrow \left\{\begin{array}{cc}1,& -1\\ i,& -i\\ -i,+i& \\ -1,& 1\end{array}\right\}$
$i,(-i)$ pair
$\begin{array}{rl}& \text{ Zero probability }=\frac{4}{{}^4{C}_1\cdot {}^4{C}_1}=\frac{1}{4}\\ & \text{ Probability (non-zero) }=1-\frac{1}{4}=\frac{3}{4}\end{array}$