This Exam P sample reference tests Bayes’ Theorem. The exact-one likelihood is binomial under each source. Combining those likelihoods with source priors gives posterior 0.09566, choice A.
Original variant: source inference after a clean inspection
A supplier provides 40% of electronic batches and has a 20% per-unit defect rate; all other suppliers provide 60% and have a 5% defect rate. Ten independent units from a batch are inspected and none is defective. Find the probability that the batch came from the first supplier.
A 0.04000
B 0.07353
C 0.09500
D 0.10679
E 0.40000
Variant answer in brief
The no-defect likelihoods are 0.8¹⁰ and 0.95¹⁰. Combining them with the source priors gives posterior 0.10679, choice D.
Setup
Setup
For each supplier class, independence makes the likelihood of ten clean inspections the tenth power of its per-unit nondefect probability.
L1=(0.80)10,L0=(0.95)10
Model
Model
Multiply the two clean-inspection likelihoods by the supplier priors and normalize the first supplier's weight.
Pr(S1∣0 defects)=0.40L1+0.60L00.40L1
Compute
Compute
The prior-weighted likelihood ratio gives posterior probability 0.106789 for the first supplier.
Pr(S1∣0)=0.106789
Answer
Answer
Given ten nondefective inspected units, the first-supplier probability is 0.10679, which is choice D.
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