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Assume that the probability that a jury in a criminal case will arrive at the correct verdict of guilty or not guilty is 0.95. Further, suppose the local police force is diligent in its work and that 99% of those brought to trial are actually guilty. Given that the jury finds the defendant guilty, find the probability the defendant is in fact guilty.

  • I tried doing a tree diagram for this problem because I thought it could really help but I got confused. I have criminal cases split into 99% guilty and 1% not guilty. Then I split the 99% guilty into .95 actually guilty and .05 not guilty. Finally the 1% not guilty into .05 actually guilty and .95% not guilty. Considering my diagram was correct the correct answer should be (0.99*0.95)/(0.99)(0.95)+(0.01)(0.05) = 1881/1882. However I believe my diagram is incorrect and therefore so is my answer. Any help would be much appreciated.
Sam
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$$ \begin{array}{ccc} 0.99 & & 0.01 \\ \text{guilty} & & \text{not guilty} \\ \overbrace{\begin{array}{ccc} \text{acquitted} & & \text{convicted} \\ 0.05 & \quad & 0.95 \\[15pt] 0.99\times0.05 & & 0.99\times 0.95 \\ =0.0495 & & = 0.9405 \end{array}} & & \overbrace{\begin{array}{ccc} \text{acquitted} & \quad & \text{convicted} \\ 0.95 & & 0.05 \\[15pt] 0.01\times0.95 & & 0.01\times0.05 \\ = 0.0095 & & =0.0005 \end{array}} \end{array} $$ So the probability of being convicted is $0.9405 + 0.0005 = 0.941.$ The probability of being in fact guilty and being convicted is $0.9405.$ Therefore $$ \Pr(\text{in fact guilty} \mid \text{convicted} ) = \frac{0.9405}{0.9405 + 0.0005} = \frac{9405}{9410} = \frac{1881}{1882}. $$

Michael Hardy
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