Question

If $A$ and $B$ are two events that the value of the determinant chosen at random from all the determinants of order $2$ with entries $0$ or $1$ only is positive and negative respectively then:

• A. $P\left(A\right)\ge P\left(B\right)$
• B. $P\left(A\right)\le P\left(B\right)$
• C. $P\left(A\right)=P\left(B\right)=\frac{1}{2}$
• D. none of the above

Answer 1

Answer: (A), (B)

$P\left(A\right)\ge P\left(B\right)$
$P\left(A\right)\le P\left(B\right)$

Since each element of a $2\times 2$ matrix can be 0 or 1.

Therefore the total number of matrices with entries 0 or 1 is $2^4=16$

Out of these 16 determinants, there are 3 positive and 3 negative.

$\therefore P\left(A\right)=P\left(B\right)=\frac{3}{16}\ne \frac{1}{2}$