Mathematics - Probability Question with Solution | TestHub

Given,

$S=\left\{M=\left[a_{ij}\right], a_{ij}\in \left\{0,1,2\right\}, \left\{1\le i,j\le 2\right\}\right\}$ 

So, let $M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ where $a,b,c,d\in \left\{0,1,2\right\}$

So, total sample space will be $3^4=81$

Now we will find $P\left(\overline{A}\right)$ or possibilities for $\left|M\right|=0$

So, for $\left|M\right|=0$ cases will be,

1.$\left[\begin{array}{ll}1& 1\\ 1& 1\end{array}\right]$ or $\left[\begin{array}{ll}0& 0\\ 0& 0\end{array}\right]$ or $\left[\begin{array}{ll}2& 2\\ 2& 2\end{array}\right]\to$ Total matrix $=3$
2. Two $1\text{'}s$ and Two $0\text{'}s$ $\to$ Total matrix $=4$
3. Two $2\text{'}s$ and Two $0\text{'}s$ $\to$ Total matrix $=4$
4. Two $1\text{'}s$ and Two $2\text{'}s$ $\to$ Total matrix$=4$
5. One $1$ and three $0 \text{'}s$ $\to$ Total matrix $=4$
6. One $2$ and Three $0\text{'}s$ $\to$ Total matrix $=4$
7. One $1$ and one $2$ and two $0\text{'}s$ $\to$ Total matrix $=8$

So, total cases for which $\left|M\right|=0$ is $31$,

So probability will be $P\left(\overline{A}\right)=\frac{31}{81}$
Hence, $P\left(A\right)=1-\frac{31}{81}=\frac{50}{81}$