You and your friend Bill enter a raffle fund-raiser together, each paying half the cost of a ticket. As a result, you and Bill win fifth prize, which is a Mani-Pedi. A Mani-Pedi is hard to split 50-50, so your friend calls you on the phone and offers to flip you for it. You call heads, he flips the coin and tells you it’s tails. Your friend is known to tell the truth 2/3 of the time and lie 1/3 of the time so you think that the coin might have landed on heads. You ask your friend to put his wife on the phone so she can report the result to the flip to you. His wife tells you that it is tails. However, his wife also is known to tell the truth 2/3 of the time and lie 1/3 of the time.
a) What is the probability that the coin is actually tails?
b) What if, instead of talking to his wife directly, Bill privately asked his wife for the result of the coin toss? She responds to Bill and then Bill reports her answer to you over the phone by saying, “She said tails.” So, Bill tells you it is tails and then he tells you that his wife said it was tails. When this occurs, what is the probability that the coin is actually tails?