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?

Answers

Solution

1/5. You can check the solution.

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: P(A|B)·P(B) = P(AB).

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 1/3·1/3 = 1/9: P(AB) = 1/9.

The second fellow might have made his claim provided both of them either told truth or both lied, which means that P(B) = 1/3·1/3 + 2/3·2/3 = 5/9. From here, P(A|B) = (1/9)/(5/9) = 1/5.

References

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

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