Four (mostly) liars

Logic Level 2

A man has four cousins. He knows that each of them, independently, has a 1/3 chance of telling the truth when making any given statement.

One of the cousins, A, makes a statement that the man and everyone else in the room hears. He doesn't know whether it's true or false.

The man asks the second cousin, B, whether A was lying. B makes a statement that the third cousin, C, and the fourth cousin, D, hear, but the man does not.

The man asks C whether B was telling the truth. C replies in a low tone and D hears his answer, but the man does not.

The man asks D whether C was telling the truth. D replies clearly, "C said that B was lying when he claimed that A was a liar."

What is the probability that A was telling the truth?

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