Question

If $a_{1},a_{2},b_{1}$ and $b_{2}$ take values in the set $\left\{1 , - 1,0\right\},$ then the probability that the equation $a_{1}a_{2}=b_{1}b_{2}$ is satisfied is $\frac{p}{q},$ ( $p$ & $q$ are co-prime) then, $q-2p$ is

Answer 1

Total ways is $3^{4}$
Now, $a_{1}a_{2}=b_{1}b_{2}$
If,
Case $\left(1\right)a_{1}a_{2}=b_{1}b_{2}=0$
If at least one of $\left(a_{1 } o r a_{2}\right)$ and $\left(b_{1} o r b_{2}\right)$ is “ $0$ ” i.e. $\left(3^{2} - 2^{2}\right)$ ways
Case $\left(2\right)$ $a_{1}a_{2}=b_{1}b_{2}=+1$
Both $a_{1},a_{2}is+1$
or $-1$
i.e. $2^{2}$ ways
Case $\left(3\right)$ $a_{1}a_{2}=b_{1}b_{2}=-1$
$2^{2}$ ways
Hence, required probability
$=\frac{\left(3^{2} - 2^{2}\right) + 4 + 4}{3^{4}}$ $=\frac{33}{81}=\frac{11}{27}$