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.

98 Questions

52 Easy

25 Medium

21 Hard

2026 Syllabus

Sample Questions

Question 1 Easy

P(A) = 0.6, P(B|A) = 0.3, P(B|A') = 0.5. Calculate P(B).

Solution

C is correct.

P(B)

= P(B \mid A) P(A) + P(B \mid A') P(A') = 0.3(0.6) + 0.5(0.4) = 0.18 + 0.20 = 0.38.

Question 2 Medium

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.

Solution

D is correct.

P(\text{2nd red}) = \frac{6}{11} \cdot \frac{7}{12} + \frac{7}{11} \cdot \frac{5}{12} = \frac{42+35}{132} = \frac{77}{132} = \frac{7}{12}

Question 3 Hard

Let

X_1, X_2, X_3

be iid Uniform

[0, \theta]

random variables, and define

M = \max(X_1, X_2, X_3)

. The parameter

\theta

has a prior distribution Uniform on

[6, 24]

. Given that

M = 6

, calculate

P(\theta > 12 \mid M = 6)

Solution

B is correct.

Step 1: Likelihood. For 3 iid Uniform

[0,\theta]

, the PDF of

M

f_{M|\Theta}(m|\theta) = 3m^2/\theta^3

. At

m = 6

L(\theta) = 108/\theta^3

for

\theta \geq 6

.

Step 2: Posterior. With uniform prior on

[6, 24]

f_{\Theta|M}(\theta|6) \propto \theta^{-3} \text{ for } \theta \in [6, 24]

Step 3: Normalizing constant.

\int_6^{24} \theta^{-3}\,d\theta = \left[-\frac{1}{2\theta^2}\right]_6^{24} = -\frac{1}{1152} + \frac{1}{72} = \frac{15}{1152} = \frac{5}{384}

Step 4: Numerator.

\int_{12}^{24} \theta^{-3}\,d\theta = -\frac{1}{1152} + \frac{1}{288} = \frac{3}{1152} = \frac{1}{384}

Step 5:

P(\theta > 12 \mid M = 6) = \frac{1/384}{5/384} = 1/5 = 0.200

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Free SOA Exam P (Probability) Conditional Probability & Bayes Practice Questions

Sample Questions

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