An man was found dead with knife wounds in the biggest family gathering in the world with 4,000 relatives partying in a mansion. The home was put under lockdown so that no guests will leave and fingerprint analysis of the murder weapon using a rapid test kit was positive on the 400th person tested. Assuming that an innocent person has only a 1 in 500 chance of a false positive result, what is the probability that this person is guilty? Select the correct answer below.

Question

BlackTom AI · Accepted Answer

We need to evaluate the probability that the person is guilty given a positive test result, using Bayes’ theorem and the information provided.
Option 1: 0.95
This would require a very high post-test probability, implying either an extremely high prior probability of guilt or an almost perfect test. The prior that any single person is the killer is 1 in 4000 (since there are 4000 relatives and one killer), which is 0.00025. The given false-positive rate is 1 in 500, i.e., 0.002. With such a low prior and a nonzero false-positive rate, the post-test probability cannot reach as high as 0.95; the evidence from one positive test would be overwhelmed by the chance of a false positive.
Option 2: 0.002
This value would reflect the false-positive rate itself, as if a positive test didn’t increase the belief at all about guilt. But because there is a real (albeit small) prior chance of guilt, and the test is positive, the probability should be higher than the false-positive rate alone, assuming the test has at least some sensitivity. Therefore 0.002 underestimates the post-test probability.
Option 3: 0.125
This corresponds to 1/8, which would require a ratio of the true-positives to false-positives that significantly boosts the posterior probability beyond the false-positive rate but without reaching the level produced by Bayes’ calculation here. It’s not aligned with the given numbers when applying the standard Bayes formula with P(G)=0.00025 and P(FP)=0.002.
Option 4: 0.250
A quarter post-test probability would imply a much stronger prior or an extremely low false-positive rate, neither of which matches the given data. With a prior of 0.00025 and FP=0.002, the posterior is driven toward a value around 0.111, not 0.25.
Option 5: 0.111
This is the result you get when you apply Bayes’ theorem with the assumptions: P(G)=1/4000=0.00025, P(Pos|G)=1 (perfect sensitivity, as not otherwise stated), and P(Pos|not G)=0.002 (false positive rate = 1/500). The denominator is 0.00025 (true positives) plus 0.99975×0.002 ≈ 0.0019995 (false positives), totaling ≈0.0022495. The ratio 0.00025/0.0022495 ≈ 0.1111, i.e., about 11.1%. This matches the Bayesian calculation given the data.
Across all options, only the last one aligns with the Bayes calculation using the stated prior and false-positive rate, yielding roughly 0.111. The other choices either misinterpret the role of the prior, misapply the false-positive rate, or disregard the impact of combining prior probability with test performance.

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