Consider the system of equations px + qy = 0, rx + sy = 0 where  $p,q,r,s\in \{-1,1\}$
Statement-1: The probability that the system of equation has a unique solution is  $\frac{1}{2}$.
Statement-2: The probability that the system of equations no solution is zero.
 

We have

$S=\{\left|\begin{array}{cc}1& -1\\ 1& 1\end{array}\right|,\left|\begin{array}{cc}1& 1\\ -1& 1\end{array}\right|,\left|\begin{array}{cc}-1& 1\\ -1& -1\end{array}\right|,\left|\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right|,\left|\begin{array}{cc}-1& 1\\ 1& 1\end{array}\right|,\left|\begin{array}{cc}1& 1\\ 1& -1\end{array}\right|,\left|\begin{array}{cc}1& -1\\ -1& -1\end{array}\right|,\left|\begin{array}{cc}-1& -1\\ -1& 1\end{array}\right|,\left|\begin{array}{cc}1& 1\\ -1& -1\end{array}\right|,\left|\begin{array}{cc}-1& -1\\ 1& 1\end{array}\right|$$\left|\begin{array}{cc}-1& 1\\ -1& 1\end{array}\right|,\left|\begin{array}{cc}1& -1\\ 1& -1\end{array}\right|,\left|\begin{array}{cc}1& 1\\ 1& 1\end{array}\right|,\left|\begin{array}{cc}1& -1\\ -1& 1\end{array}\right|,\left|\begin{array}{cc}-1& 1\\ 1& -1\end{array}\right|,\left|\begin{array}{cc}-1& -1\\ -1& -1\end{array}\right|\}$

$\therefore n\left(S\right)=16$

Now, number of favorable cases are given by ( $\Delta \ne 0$, that is either $\Delta =2or-2$)

$E=\{\left|\begin{array}{cc}1& -1\\ 1& 1\end{array}\right|,\left|\begin{array}{cc}1& 1\\ -1& 1\end{array}\right|,\left|\begin{array}{cc}-1& 1\\ -1& -1\end{array}\right|,\left|\begin{array}{cc}-1& -1\\ 1& -1\end{array}\right|,\left|\begin{array}{cc}-1& 1\\ 1& 1\end{array}\right|,\left|\begin{array}{cc}1& 1\\ 1& -1\end{array}\right|,\left|\begin{array}{cc}1& -1\\ -1& -1\end{array}\right|,\left|\begin{array}{cc}-1& -1\\ -1& 1\end{array}\right|\}$Hence probability that the system of equations has a unique solution 

$=\frac{n\left(E\right)}{n\left(S\right)}=\frac{8}{16}=\frac{1}{2}$