Question

If each element of a second order determinant is either zero or one, 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 $\frac{1}{2}$ ), is

• A. $\frac{5}{16}$
• B. $\frac{3}{16}$
• C. $\frac{7}{16}$
• D. $\frac{1}{16}$

Answer 1

Answer: (B)

A second order determinant has four entries which may be 0 or 1 .
Total number of determinants $=2^4=16$
The only positive determinants are $\begin{vmatrix}1 & 0 \\ 0 & 1\end{vmatrix},\begin{vmatrix}1 & 0 \\ 1 & 1\end{vmatrix}$ and $\begin{vmatrix}1 & 1 \\ 0 & 1\end{vmatrix}$.
Since, each entry of the above determinant can be selected with probability $\frac{1}{2}$, therefore
Required probability $=3\left(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\right)=\frac{3}{16}$.
Alternate Method
Here, total number of cases $=16$
Favourable number of cases $=3$
$\therefore$ Required probability $=\frac{3}{16}$