This Exam P sample reference tests Bayes’ Theorem. Survival weights for the three arrival classes sum to 0.924; the serious-class weight is 0.27, so the posterior is 0.29221, choice B.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 0.06 is the critical-class survival weight 0.10(0.60), not the serious-class posterior.
CThe value 0.30 is the prior serious-class share. Conditioning on survival changes that share because survival rates differ.
EThe value 0.64 is 0.594/0.924 to rounding, the posterior probability of the third arrival class among survivors.
Original practice · fully worked
Original variant: posterior source after a mixed two-stage outcome
Thirty of every hundred processes run in mode A; all others use mode B. An individual trial succeeds with probability 0.80 in mode A and 0.20 in mode B. Two conditionally independent trials produce one success followed by one failure. Find the posterior probability of mode A.
A 0.160
B 0.200
C 0.300
D 0.500
E 0.800
Variant answer in brief
Both modes assign likelihood 0.16 to the ordered mixed outcome, so the evidence does not change the prior. The posterior remains 0.30, choice C.
Setup
Setup
Compute the ordered success-then-failure likelihood under each process mode.
LA=0.80(0.20)=0.16
LB=0.20(0.80)=0.16
Model
Model
Apply Bayes' formula by multiplying each equal likelihood by its mode prior and normalizing the mode-A weight.
Pr(A∣SF)=0.30LA+0.70LB0.30LA
Compute
Compute
Both likelihoods equal 0.16, so they cancel from the Bayes ratio; the posterior remains the prior 0.30.
Pr(A∣SF)=0.048/(0.048+0.112)=0.30
Answer
Answer
The posterior probability of mode A is 0.300, which is choice C.
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.
