In a pool of polyhedral dice (made up of $4$ to $6$ $2$-sided, $4$-sided, $6$-sided, $8$-sided, $10$-sided, and $12$-sided dice) how do you determine the chances of matching the last digit rolled on a $20$-sided die?
I am interested in determining the chances of one or more of the pool results matching a simultaneously-thrown d$20$'s final digit. (i.e. a "$1$" on a die matching the "$1$" or "$11$" on the d$20$, or a "$10$" on a d$12$ matching the "$10$" or "$20$" on the d$20$)
Specific points of interest for me include:
- What are the chances of $0, 1, 2, \dots$ matches?
- Does an increase in the number of sides of the dice in the pool translate into greater chances of matches?
- Is there a difference between a d$10$ and a d$12$ in terms of how likely one or the other is to generate a match to the d$20$?