## Monday, January 07, 2008

### Truth, Lies, and Islands

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?

1. 1/3 I think. It doesn't matter whether the second fellow stepping forward and confirming the statement was telling the truth or not.

2. 1/9? if the first fellow was telling the truth, then the second fellow must have been telling the truth about the truth the first was telling. however if the first fellow lied, then the second fellow was as well.

3. If we re-write the question: what was the probability the first one told the truth given that the second one said it was so.

Assuming the two islanders statements are independent, the probability they both told the truth is 1/3 * 1/3 = 1/9.

The probability of the second guy backing up the first statement is the probability they both lied or they both told the truth = 1/3 * 1/3 + 2/3 * 2/3 = 5/9.

So the probability of the first guy telling the truth, given that the second guy claiming it was true = P(A|B) = P(AB)/P(B) = (1/9) / (5/9) = 1/5.

After looking at the question, I can understand the confusion. I probably should have written it more clearly.

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