Due on Sep 21 (50 points)

1. Three prisoners, A, B, and C with apparently equally good records have applied for parole. The parole broad has described to release two of the three, and the prisoners know this but not which two. A warder friend of prisoner A knows who are to be released. Thinking that it is immoral to let A know if he will be released on not. He only tells him that C is going to be released. What is the probability that B is to be released?

2. A 1 foot long stick is broken randomly in three parts. Show that the expected length of the shortest piece has length 1/9.

3. With two type of indistinguishable coins, type-A coin is heavily biased towards head with P(H)=0.8, and type-B coin heavily biased towards tail with P(H)=0.2. There are 1 type-A coins and 99 type-B coins in a bag. A player randomly draws a coin, tosses the coin two times and get heads for both times. Estimate the probability of getting another head for the third time using a) ML, b) MAP, and c) Bayesian estimation.

4. For the same setting in the last problem, use Lea to estimate the probability of getting another head using a) ML, b) MAP, and c) Bayesian estimation after observing 8 heads out of 10 tosses.

5. Repeat the problem above of estimating the probability of 11th head after observing 8 heads out of 10 tosses but assuming that we don't have two types of coins but the prior probability of head is Beta-distributed with parameters \(a\) and \(b\) as 2 and 3.

6. (Extra-credit) With the same number of type-A and type-B coins in the bag, at least how many times we need to toss the coin until that we can tell the type of the coin with 99% certainty?