Question

A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the determinant is positive is

• A. $\frac{3}{16}$
• B. $\frac{3}{8}$
• C. $\frac{5}{8}$
• D. $\frac{7}{8}$

Answer 1

The correct option is A $\frac{3}{16}$

A determinant of order $2$ can be filled with digits $0$ or $1$ in $2^4=16$ ways.
A positive determinant can be made in the following $3$ ways:
$\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|,\left|\begin{array}{cc}1& 0\\ 1& 1\end{array}\right|,\left|\begin{array}{cc}1& 1\\ 0& 1\end{array}\right|$
Hence, probability = $\frac{3}{16}$