Four witnesses A, B, C, D at a trial each speak the truth with probability 1/3 independently of each other. In their testimonies, A claimed that B denied that C declared that D lied. What is the (conditional) probability that D told the truth?
Note: I was told that I can assume, say taking the statement C declared that D lied to mean if C told the truth then D lied but if C lied then D told the truth.
I started by listing out the sample space, but obviously with four witnesses that got large very quickly... So I decided to just list the ones that pertain to the situation described. Here is what I have so far:
(1) D lied, C truth, B lied, A truth
(2) D truth, C lied, B truth, A truth
(3) D lied, C truth, B lied, A lied
(4) D truth, C lied, B truth, A lied
But I don't think these are all the possible sequences. Could someone put me on the right track? I'm getting stumped by the part "B denied"