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Probability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSProbability3 Marks
Question
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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Answer

There are four entries in determinant of $2 \times 2$ order.
Each entry may be filled up in two ways with $0$ or $1.$
$\therefore$ number of determinants that can be formed $= 2^4 = 16$
$\therefore$ total number of cases $= 16$
The value of determinant is positive in the cases
$ \begin{vmatrix} 1\ 0\\0\ 1 \end{vmatrix}, \begin{vmatrix} 1\ 0\\1\ 1 \end{vmatrix}, \begin{vmatrix} 1\ 1\\0\ 1 \end{vmatrix},$
$\therefore$ number of favourable cases $= 3$
$\therefore$ the probability that the determinant is positive $=\frac{3}{16}.$

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