Ana and Banana are rolling a standard six-sided die. Ana rolls the die twice, obtaining $a_{1}$ and $a_{2}$, then Banana rolls the die twice, obtaining $b_{1}$ and $b_{2}$. After Ana's two rolls but before Banana's two rolls, they compute the probability $p$ that $a_{1} b_{1}+a_{2} b_{2}$ will be a multiple of 6 . What is the probability that $p=\frac{1}{6}$ ?
Solution: If either $a_{1}$ or $a_{2}$ is relatively prime to 6 , then $p=\frac{1}{6}$. If one of them is a multiple of 2 but not 6 , while the other is a multiple of 3 but not 6 , we also have $p=\frac{1}{6}$. In other words, $p=\frac{1}{6}$ if $\operatorname{gcd}\left(a_{1}, a_{2}\right)$ is coprime to 6 , and otherwise $p \neq \frac{1}{6}$. The probability that $p=\frac{1}{6}$ is $\frac{\left(3^{2}-1\right)\left(2^{2}-1\right)}{6^{2}}=\frac{2}{3}$ where $\frac{q^{2}-1}{q^{2}}$ corresponds to the probability that at least one of $a_{1}$ and $a_{2}$ is not divisible by $q$ for $q=2,3$. $\fbox{\frac{2}{3}}$.