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

Answer: 5

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 $\left(a_{1}or{a}_2\right)$ and $\left(b_{1}or{b}_2\right)$ 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
$=\frac{\left(3^2-2^2\right)+4+4}{3^4}$ $=\frac{33}{81}=\frac{11}{27}$