Say I have 8 fair dice and one trick die that always rolls "1."
I pick one die out of the nine and roll it three times, getting three 1's in a row.
What is the probability the dice I picked was fair?
My thought process:
I want to use conditional probability in the following way:
$P(F | 3x1)$ = $P(F \cap3x1)/P(3x1) $.
I know that if I roll a fair die 3 times, there are $6^3$ possibilities, but if I roll the trick die three times, there is only one possibility. How do I put these ideas together?