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I have this question for a test: A judge is 35% sure that Tim commited burglary. The witness would lie at a probability of 0.25 if Tim is guilty but would tell the truth if Tim is innocent What probability would the witness commit perjury? Is P(witness commit perjury) = 0.35*0.25 = 0.0875 ?

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    $\begingroup$ No I didn't! My 0.3$: The fact that judge is sure about something has nothing to do with (a) a witness, (b) the truth - doesn't it..? However this is his prior expectation. $\endgroup$ – Tim Jan 29 '15 at 9:22
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Yes, I believe you are correct. Let $B$ represent that Tim is a burglar. We are given $P(B) = 0.35$. The probability that the witness will commit perjury, call it $P(P)$, can be split into two disjoint situations:

  1. The witness lies given Tim is innocent: $P(P|\overline{B})$ = 0
  2. The witness lies given Tim is guilty: $P(P|B)$ = 0.25

Now we know the following: $$ \begin{align} P(P) &= P(P \cap B) + P(P \cap \overline{B})\\ &= p(P|B)P(B) + p(P|\overline{B})P(\overline{B})\\ &= 0.25\times .35 + 0\times .65\\ &= .0875 \end{align} $$

Now if the question was what is the judge's posterior probability for Tim's guilt given the witness testimony, that's a different question.

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  • $\begingroup$ For me the thoughts that judge has sound rather as a prior then posterior but it could be a matter of language... $\endgroup$ – Tim Jan 29 '15 at 9:32
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    $\begingroup$ I agree, it is the prior of the Judge's belief. As you say above, it is irrelevant to the OPs question. A more interesting question, I think, would be what is the judge's posterior belief if the witness testifies. $\endgroup$ – Avraham Jan 29 '15 at 9:34

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