Three Logicians and Five Hats

A bag holds five hats: two red and three blue. Alice, Bob and Carol, all perfect and truthful logicians, each draw one hat at random without looking and put it on, so each sees the other two hats but not their own. The remaining two hats stay hidden. All of this is common knowledge. A referee then asks them in turn, Alice, Bob, Carol, Alice, Bob, Carol, and so on: "Do you know the colour of your own hat?" Every answer is heard by all, and the questioning stops at the first "yes". What is the expected number of questions asked, counting the one answered "yes"? Give your answer as a reduced fraction.

• expected value
• graph theory
• information
• optimal stopping

Source: Classic "wise men and hats" puzzle in wide circulation, here with a modified colour split and an expectation question; no traceable original source. Statement written for AxiomIQ.