Nuts come in two different chiralities (threadings); left-handed (L) and right-handed (R). Oleson's hardware store sells packs containing both kinds in one pack.

- Oleson sells 5 types of nut pack. The types are enumerated by an \(x\) running from 1 through 5.

- All packs contain 100 nuts. A pack of type \(x\) has \(10 x\) left-handed (L) nuts; the remainder are right-handed (R).

- Oleson has just not sold any of the packs, so he knows he has \(30-5 x\) packs of type \(x\) in his store.

- The packs are unfortunately unmarked, so now the entire inventory of them is now on sale. You pick a pack at random, and then pick a nut to study it. It is a left-handed nut (L). You put it back before picking a new random nut; it, too, is left-handed (L).

(a) Write down the function \(f(x)\) expressing the prior probabilities \({ }^{3}\) you picked a pack of type \(x\).

(b) Find the likelihood function \(g(x)\).

(c) Fill in the table to find the posterior probability that you picked a pack of type \(x\).

(d) You put the nut back. What is the probability that the next random nut you pick is left-handed (L)?