What is the Probability that Exactly One Pair among Three Integers is Coprime

Question

Given three positive integers $a,b,c$ it isn't difficult to show that $$ P(\gcd(a,b,c)=1) = \prod_{p}\left(1-p^{-3}\right)=\frac1{\zeta(3)} $$ where $\zeta$ is the Riemann zeta function. For the next problems I define the events $A_p=\{a\in\mathbb{N}:p|a\},$ $B_p=\{b\in\mathbb{N}:p|b\},$ $C_p=\{c\in\mathbb{N}:p|c\}$. I call $P_n$ the probability that exactly $n$ out of the three pairs $(a,b),(b,c),(a,c)$ are coprime. Using $A_p,B_p,C_p$ I can show $$ \begin{aligned} P_3 &= \prod_{p} \left(1+\frac2p\right)\left(1-\frac1p\right)^2 \\ P_2 &= \prod_{p} \left(1+\frac1p-\frac1{p^2}\right)\left(1-\frac1p\right) \end{aligned} $$ but I can't get $P_1$ or $P_0$ in a similar form. In fact, it seems like it's impossible to derive either of $P_1$ and $P_0$ from $A_p,B_p,C_p$ and their compliments. How would you derive a product definition for $P_1$ and $P_0$ like $P_3$ and $P_2$ have above?