I have a box of dice that contains a 4-sided die, a 6-sided die, an 8-sided die, a 12-sided die, and a 20-sided die. If you have ever played Dungeons & Dragons, you know what I am talking about.
Suppose I select a die from the box at random, roll it, and get a 6. What is the probability that I rolled each die?
$$P(\text{rolled a $k$-sided die } | \text{ rolled a 6}) = \dfrac{P(\text{rolled a 6 } | \text{ rolled a $k$-sided die}) \cdot P(\text{rolled a $k$-sided die})}{P(\text{rolled a 6})}$$
And $P(\text{rolled a 6}) = \dfrac{1}{5}\left(0 +\dfrac{1}{6} +\dfrac{1}{8} +\dfrac{1}{12} +\dfrac{1}{20} \right) = \dfrac{17}{200}$, meaning:
$$P(\text{rolled a $k$-sided die } | \text{ rolled a 6}) = \dfrac{\left(\dfrac{1}{k} \cdot [k \geq 6]\right) \cdot \dfrac{1}{5}}{\dfrac{17}{200}} = \dfrac{40}{17k} \cdot [k \geq 6]$$
Where $[x]$ is the Iverson bracket, which evaluates to $1$ if $x$ is true, and $0$ if $x$ is false.