Sacks with handles; the contents are white (W) and black (B) balls.

- You have seven sacks with an index \(x\) running from 1 through 7.

- Sack \(x\) has \(x\) handles, and in a random pick, each handle has equal probability of being picked.

- All the sacks are filled with white (W) and black (B) balls, a total of 50 in each sack. In sack \(x\), there are \(x^{2}\) white balls.
- You pick a random ball from your sack, and get black. You put the ball back in the sack.

(a) Write down the function \(f(x)\) expressing the prior probabilities you picked sack \(x\) when you pulled a random handle. \({ }^{2}\)

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

(c) Fill in the table to find the posterior probability that you picked sack \(x\) when you pulled a random handle.

(d) What is the probability that the observation of your next trial, " 2 samplings with replacement" yields W-W?