Browse Questions

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

$\begin{array}{1 1} \large\frac{3}{16} \\ \large\frac{5}{16} \\ \large\frac{7}{16} \\ \large\frac{9}{16} \end{array}$

Given that a second order determinant has 4 entries, which may be 0 or 1. Total number of determinants = $2^4 = 16$
We need to find the probability that the determinant is positive.
The only positive determinants are $\begin{vmatrix} 1 &0 \\ 0& 1 \end{vmatrix}, \begin{vmatrix} 1 &1 \\ 0& 1 \end{vmatrix} \;\text{and}\; \begin{vmatrix} 1 &0 \\ 1& 1 \end{vmatrix}$
Therefore the required probability = $\large\frac{3}{16}$