Question

If each element of a second order determinant is zero or one , what is the probability that the values of the determinant is positive?(Assume that the individual entries of the determinant are chosen independently, each value being assumed with probability$\frac{1}{2}$

Answer 1

A second order determinant has four entries which may be 0 or 1.

Total number of determinants=$2^{=}16$

The only positive determinants are $\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|\left|\begin{array}{cc}1& 0\\ 1& 1\end{array}\right|and\left|\begin{array}{cc}1& 1\\ 0& 1\end{array}\right|$.

Since, each entry of the above determinant can be selected with probability $\frac{1}{2}$ , therefore required probability.

$=3(\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2})=\frac{3}{16}.$

Alternatively Here, total number of cases=16

Favourable number of cases=3

$\therefore Requiredprobability=\frac{3}{16}$