Bayes’s rule can be expressed as follows:
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P(A | B) = |
P(B | A) x P(A) |
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P(B) |
For the current problem, let’s use the notation that G refers to the prior probability that the suspect is guilty (before we know anything about the lab report) and E refers to the evidence of a blood match. We want to know P(G|E). Substituting in the above, we put in G for A and E for B to obtain:
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P(G | E) = |
(P(E | G)×P(G)) |
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(P(E)) |
To compute Bayes’s rule and solve for P(G | E), it may be helpful to use a table. The values here are the same as those used in the fourfold table here.
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COMPUTATION OF BAYES’S RULE |
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Hypothesis (H) (1) |
Prior Probability P(G) (2) |
Evidence Probability P(E | G) (3) |
Product (4) = (2)(3) |
Posterior Probabilities P(G | E) (6) = (4)/Sum |
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Guilty |
.02 |
.85 |
.017 |
.104 |
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Innocent |
.98 |
.15 |
.147 |
.896 |
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Sum = .164 = P(D) |
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Then, rounding, P(Guilty | Evidence ) = .10
P(Innocent | Evidence) = .90