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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 $\frac{1}{2}$ ).
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There are four entries in determinant of $2 \times 2$ order. Each entry may be filled up in two ways with 0 or 1
$\therefore \quad$ Number of determinants that can be formed $=2^4=16$ The value of determinant is positive in the cases
$$
\left|\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right|,\left|\begin{array}{ll}
1 & 0 \\
1 & 1
\end{array}\right|,\left|\begin{array}{ll}
1 & 1 \\
0 & 1
\end{array}\right|
$$
The probability that the determinant is positive $=\frac{3}{16}$
$\therefore \quad$ Number of determinants that can be formed $=2^4=16$ The value of determinant is positive in the cases
$$
\left|\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right|,\left|\begin{array}{ll}
1 & 0 \\
1 & 1
\end{array}\right|,\left|\begin{array}{ll}
1 & 1 \\
0 & 1
\end{array}\right|
$$
The probability that the determinant is positive $=\frac{3}{16}$
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