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 non-negative ?

Answer 1

In a $2\times 2$ determinant there are 4 elements and each element can take 2 values. So, total number of $2\times 2$ determinants with elements 0 and 1 is $2^4=16$. Out of these determinants the values of the following determinant are negative :

$\left|\begin{array}{cc}1& 1\\ 1& 0\end{array}\right|,\left|\begin{array}{cc}0& 1\\ 1& 1\end{array}\right|,\left|\begin{array}{cc}0& 1\\ 1& 0\end{array}\right|$.

Hence, required probability = $\frac{3}{16}$