Question

Let $S$ be the sample space of all $3\times 3$ matrices with entries from the set {0,1}. Let the events $E_1$ and $E_2$ be given by
$E_1=\{A\in S:detA=0\}$ and
$E_2=${A $\in S:$ sum of entries of $A$ is $7$}
If the matrix is chosen at random from $S,$ then the conditional probability $P\left(E_1|E_2\right)$ equals

Answer 1

Sample space $S=2^9$
$P\left(E_1|E_2\right)=\frac{P(E_1\cap E_2)}{P\left(E_2\right)}$
$E_2:$Sum of entries of A is 7 i.e matrix consist of 7 ones and 2 zeros
such as $\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 0& 0\end{array}\right|$
$\therefore$ $n\left(E_2\right)=\frac{9!}{7!2!}=\frac{8\times 9}{2}=36$
For $detA$ to be zero, both zero should be in same row or same column
Therefore, for selecting 2 zero in same row from 3 row matrix $=3{\times }^3{C}_2=3\times 3=9$
Similary for selecting 2 zero in same column from 3 column matrix$=3{\times }^3{C}_2=3\times 3=9$
So, $n(E_1\cap E_2)=9+9=18$
$P\left(E_1|E_2\right)=\frac{P(E_1\cap E_2)}{P\left(E_2\right)}=\frac{18}{36}=\frac{1}{2}=0.5$