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 $\cfrac{1}{2}$ ).

.: Determinants of the required type are :
$\left| \begin{array}{l}1\,\,\,\,\,\,\,0\\0\,\,\,\,\,\,\,1\end{array} \right|,\left| \begin{array}{l}1\,\,\,\,\,\,\,0\\1\,\,\,\,\,\,\,1\end{array} \right|$ and $\left| \begin{array}{l}1\,\,\,\,\,\,\,1\\0\,\,\,\,\,\,\,1\end{array} \right|$

Since each entry of the said determinant can be selected with equal probability,

therefore
Required probability $= 3\left( {\cfrac{1}{2} \times \cfrac{1}{2} \times \cfrac{1}{2} \times \cfrac{1}{2}} \right) = \cfrac{3}{{16}}$