Bayes update on a witness naming a trader

Question

Confidential IP has leaked from a quant firm. Two thirds of the staff are traders and one third are researchers. A witness, who is right $2/3$ of the time, says the leaker was a trader. Given this testimony, what is the probability a trader really was responsible?

Short answer

$4/5$ . By Bayes, $\dfrac{\tfrac23\cdot\tfrac23}{\tfrac23\cdot\tfrac23+\tfrac13\cdot\tfrac13}=\dfrac{4/9}{5/9}=\tfrac45$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Conditional Probability, Total Probability, and Bayes' Rule
Worked Examples — Probability Trees, Monty Hall, and Bayesian Updating
Sample Spaces, Events, and the Probability Axioms

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