This Exam P sample reference tests Conditional Probability. The observed mixed pair has probability 3/8 under replacement and 1/2 without replacement. Equal prior odds and Bayes' rule give a posterior probability of 3/7, so choice C is correct.
How to solve this Conditional Probability question
Setup
Setup
Let B and A denote the two possible selectors, and let M denote observing one assignment of each type. Alternating workdays give equal prior probabilities.
Pr(B)=Pr(A)=21
Model
Model
Calculate the mixed-pair likelihood under each selection protocol, allowing either order.
Pr(M∣B)=2(41)(43)=83
Pr(M∣A)=4133+4331=21
Compute
Compute
Apply Bayes' rule with the two equal prior weights.
Pr(B∣M)=(1/2)(3/8)+(1/2)(1/2)(1/2)(3/8)
Pr(B∣M)=3/8+1/23/8=73
Answer
Answer
The posterior probability that the replacement protocol generated the pair is 3/7.
73(C)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThis computes only the joint numerator P(B and M)=(1/2)(3/8)=3/16 and omits division by P(M).
BThis reports the likelihood P(M | B)=3/8, reversing the desired conditional probability.
DThis reports the marginal evidence probability P(M)=(1/2)(3/8)+(1/2)(1/2)=7/16.
EThis leaves the equal prior P(B)=1/2 unchanged and therefore ignores the different likelihoods 3/8 and 1/2.
Original practice · fully worked
Original variant: archived anomaly attribution
A reliability archive contains 120 reports from production line A and 80 reports from line B. Reviewers marked 30 line-A reports and 40 line-B reports for escalation. One marked report is selected uniformly for a root-cause drill. Calculate the probability that it came from line B.
A 1/4
B 7/20
C 2/5
D 3/7
E 4/7
Variant answer in brief
Among the 70 marked reports, 40 came from line B. The requested probability is therefore 40/70=4/7, so choice E is correct.
Setup
Setup
Restrict the archive to reports eligible for the root-cause selection.
nmarked=30+40=70
nB,marked=40
Model
Model
Uniform selection from the marked subset makes the posterior source probability a ratio of counts within that subset.
Pr(B∣marked)=nmarkednB,marked
Compute
Compute
Insert the two marked-report counts and reduce the fraction.
Pr(B∣marked)=30+4040
Pr(B∣marked)=74=0.571428…
Answer
Answer
The probability that the selected marked report came from line B is 4/7.
The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.
