Consider a die-spinner model. The first die has one "marked" face and five "unmarked" faces, whereas the second die has four "marked" faces and two "unmarked" faces. There are three spinners, each with five equally spaced sectors marked 3 or 8. The first spinner has one sector marked 3 and four marked 8, the second has two marked 3 and three marked 8, and the third has four marked 3 and one marked 8. One die and one spinner are selected at random. If rolling the die produces an unmarked face, no claim occurs. If a marked face occurs, there is a claim and then the spinner is spun once to determine the amount of the claim. Hint: treat a combination of die and spinner as Φ. 

(a) Determine the Bayesian premium E (X₂ | X₁ = 3). 

(b) Determine the Bühlmann credibility premium to approximate E (X₂ | X₁ = 3).