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 value of the determinant chosen is positive is $\frac{k}{16}$. Find the value of $k$.

Answer 1

$n=$ total no. of ways $=2^4=16,$
since each of the four places in a determinant of order 2 can be filled in two ways by 0 or 1,
$m=$ favourable no, of ways $=3$ since the value of the determinant is positive when it is
$\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|$ or $\left|\begin{array}{cc}1& 0\\ 1& 1\end{array}\right|$ or $\left|\begin{array}{cc}1& 1\\ 0& 1\end{array}\right|.$
Hence $P=\frac{m}{n}=\frac{3}{16}.$