# Unreliable Witness

The inhabitants of an island tell truth one third of the time. They lie with the probability of 2/3.

On an occasion, after one of them made a statement, another fellow stepped forward and declared the statement true.

What is the probability that it was indeed true?

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Copyright © 1996-2018 Alexander Bogomolny

1/5. You can check the solution.

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Copyright © 1996-2018 Alexander Bogomolny

The probability we are looking for is the conditional probability P(A|B) of the first fellow's statement being true (event A) provided the second fellow claims that it is (event B) indeed so.

Let's examine the other two probabilities in the standard definition:

AB is the concurrent event of the statement being true and the second fellow saying so, which only happens when both of them tell the truth. The probability of this event is

The second fellow might have made his claim provided both of them either told truth or both lied, which means that

### References

- R. Falk,
*Understanding Probability and Statistics*, A K Peters, 1993

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