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Probability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSProbability1 Mark
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 a 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 $= 2^4 = 16$
The value of determinants is positive in the following cases:
$\left| {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right|,\left| {\begin{array}{*{20}{c}} 1&0 \\ 1&1 \end{array}} \right|,\left| {\begin{array}{*{20}{c}} 1&1 \\ 0&1 \end{array}} \right| = 3$
Therefore, the probability that the determinant is positive $ = \frac{3}{{16}}$

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