Verified solution · Exam P
This Exam P sample reference tests Bayes' Theorem. Normalizing the three proportional category probabilities gives priors 0.5, 0.4, and 0.1. Their zero-count contributions are 0.05, 0.08, and 0.04, so the requested posterior is 0.04/0.17=0.235294 and choice C.
Independent solution
Setup
Normalize the category probabilities from their stated ratios.
Model
Multiply each category prior by its conditional zero-count probability.
Compute
Normalize category C's contribution by the total probability of zero claims.
Answer
The conditional probability rounds to 0.235.
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Original practice · fully worked
A signal is generated in mode R, S, or T with prior probabilities 0.45, 0.35, and 0.20. Conditional on the mode, its score Z lies between 0 and 1 with densities f_R(z)=2z, f_S(z)=2(1-z), and f_T(z)=1. A score of 0.80 is observed. Calculate the posterior probability that the signal used mode T.
At the observed score, the prior-weighted density contributions of modes R, S, and T are 0.72, 0.14, and 0.20. Normalizing T's contribution gives 0.20/1.06=0.188679, so choice B.
Setup
Evaluate each conditional density at the observed score.
Model
For a continuous observation, Bayes' theorem uses likelihood densities. Multiply each density by its mode prior.
Compute
Normalize mode T's contribution by the mixture density at the observation.
Answer
The posterior probability rounds to 0.189.
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Source reference: official SOA sample-question PDF. Last reviewed 2026-07-31.
