Say I have two dice, an $a$-sided die and a $b$-sided. For this example, let's say Die A has 6 sides while Die B has 8. If I were to roll both dice at the same time, what is the probability that I would roll a 1 on at least one of the dice?
Generally, how would this work if Die A had $a$ sides and Die B had $b$ sides? Additionally, how would this work if I added a third die of $c$ sides?
$$P(\sim \!\!1 \textrm{ on both}) = \frac{a-1}{a} \cdot \frac{b-1}{b}$$
$$P(1 \textrm{ on at least one}) = 1-\frac{a-1}{a} \cdot \frac{b-1}{b}$$
For your example ($a=6$ and $b=8$)
$$P(1 \textrm{ on at least one}) = 1-\frac{5}{6}\cdot \frac{7}{8}=1-\frac{35}{48}=\frac{13}{48}.$$
With three dice:
$$P(1 \textrm{ on at least one}) = 1-\frac{a-1}{a} \cdot \frac{b-1}{b} \cdot \frac{c-1}{c}$$