Four (mostly) liars
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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1 solution
Break it down into cases.
A lying (2/3) A truthful (1/3)
A lying, B says A lied (2/9) A lying, B said A truthful (4/9) A truthful, B said A lying (2/9) A truthful, B said A truthful (1/9)
The next phase has 8 cases, and the final phase 16. Trace back up to the ones that meet the conditions, and you'll find 41 cases, of which A is telling the truth 13 times.