A bag contains four dice: a 4-sided die, two 8-sided dice and a 12-sided die. A person chooses a die from the bag at random and rolls it. The number of sides that the selected die has is denoted by the random variable and the number rolled by the random variable .

1. Derive the probability distribution of the number rolled with the selected die. Use your expression for to determine the probability that the first number rolled is 3. That is, what is ?
2. Given that the roll was a 3, what's the posterior probability that the selected die had 12 sides? That is, what is ?
3. Suppose the person rolls the same die a second time. Denote the random variable of this second roll by . If the second roll comes up a 7, how does this change the posterior probability that the selected die has 12 sides? Specifically, what is ?
4. Suppose the person rolls the same die a third time after telling you that the first two rolls were 3 and 7. Denote the third roll by . What is the predictive probability distribution of seeing 2 on this final roll given the knowledge of the first two rolls? I.E., what is ?