Free SOA Exam P (Probability) Conditional Probability & Bayes Practice Questions
Conditional probability and Bayes' theorem are among the most frequently tested topics on SOA Exam P. Questions require calculating posterior probabilities, applying the law of total probability, and combining multiple conditioning events in multi-step problems.
An urn contains 7 red and 5 blue balls. Three balls are drawn without replacement. Verify P(2nd ball is red) = 7/12 using the law of total probability conditioning on the first draw.
Let X1,X2,X3 be iid Uniform[0,θ] random variables, and define M=max(X1,X2,X3). The parameter θ has a prior distribution Uniform on [6,24]. Given that M=6, calculate P(θ>12∣M=6).
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Correct Answer: B
Solution
B is correct.
Step 1: Likelihood. For 3 iid Uniform[0,θ], the PDF of M is fM∣Θ(m∣θ)=3m2/θ3. At m=6: L(θ)=108/θ3 for θ≥6.
Step 2: Posterior. With uniform prior on [6,24]: fΘ∣M(θ∣6)∝θ−3 for θ∈[6,24]
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