Is it correct to setup the sample space made of 26 outcomes like so?
outcomes from 6-sided dice: [ 1 2 3 4 5 6 outcomes from 8-sided dice: 1 2 3 4 5 6 7 8 outcomes from 12-sided dice: 1 2 3 4 5 6 7 8 9 10 11 12 ]
This works. It looks like a joint distribution of 'the type of dice' (rows) and the 'number rolled' (columns).
You can fill in the probabilities of each of the 26 samples in this space,
outcomes from 6-sided dice: [ a b c d e f
outcomes from 8-sided dice: g h i j k l m n
outcomes from 12-sided dice: o p q r s t u v w x y z ]
then consider the probability of the type of dice, conditional on the outcome being 1.
I will fill in the first row for you, e.g. the probability that the dice is 6-sided (1/3) and the roll is 1 (1/6) is the product 1/3 x 1/6 = 1/18, or with a different denominator 4/72 (72 is gonna be useful for comparison with the other cases)
outcomes from 6-sided dice: [ 4/72 4/72 4/72 4/72 4/72 4/72
outcomes from 8-sided dice: g h i j k l m n
outcomes from 12-sided dice: o p q r s t u v w x y z ]