Question:
If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability $\frac{1}{2}$ ).
Solution:
The total number of determinants of second order with each element being 0 or 1 is $(2)^{4}=16$
The value of determinant is positive in the following cases. $\left|\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right|\left|\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right|\left|\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right|$
$\therefore$ Required probability $=\frac{3}{16}$