Question

There are two coins in a sack. They look and feel identical in every way, except that one is a regular coin with a head and tail side; the other is a 2 headed coin with a head on both sides. When the regular coin is flipped, H and T each show with probability .5. When the 2 headed coin is flipped, H shows with probability 1. Each coin is equally likely to be pulled out of the sack. A coin is pulled out of the sack and flipped, and H comes up. We will determine the probability it is the 2 headed coin using Bayes’s rule.

1. (a) What is the prior probability of pulling out the 2 headed coin?

2. (b) What is the probability of observing H, given that the 2 headed coin is pulled out?

3. (c) What is the probability of observing H, given that the fair coin is pulled out?

4. (d) What is the probability the 2 headed coin was pulled out, given that H is showing on 1 flip?

Consider the same sack of coins as the previous problem. Suppose a single coin is pulled out and flipped twice. For a given coin, the flips are independent, i.e., the probability of H on one flip has no effect on the probability of H on the other flip.

1. (e) What is the probability it is the 2 headed coin, given that H shows on both flips?

2. (f) What is the probability it is the 2 headed coin, given that T shows on at least one flip? Write out Bayes’s rule.

Answer 1