Worked solution · Exam P

SOA Exam P Sample Question #408 Solution

This Exam P sample reference tests Conditional Probability. This is a Bayesian posterior-predictive calculation after two observed successes. Updating the selected model and averaging its next-trial success rate gives 0.90, which selects choice E.

Independent analysis by ActuaryProof · reviewed July 31, 2026

Official wording withheld. Read the problem in the official SOA sample-question PDF. This page provides only our independently written analysis.

Read the solution Open official source

Independent solution

How to solve this Conditional Probability question

Setup

Let F denote selection of the ordinary model and L denote selection of the model that always succeeds. The two models have equal prior probability.

\Pr(F)=\Pr(L)=\frac12

\Pr(HH\mid F)=\frac14,\qquad \Pr(HH\mid L)=1

Model

Apply Bayes' theorem to the two-success evidence, then average the next-trial success probability over the posterior model probabilities.

\Pr(L\mid HH)=\frac{(1)(1/2)}{(1/4)(1/2)+(1)(1/2)}=\frac45

\Pr(F\mid HH)=\frac15

Compute

The next trial succeeds with probability one under L and one-half under F.

\Pr(H_3\mid HH)=\left(\frac45\right)(1)+\left(\frac15\right)\left(\frac12\right)

\Pr(H_3\mid HH)=\frac9{10}=0.90

Answer

The posterior-predictive success probability is 0.90.

\boxed{\frac9{10}\quad\text{(E)}}

Test the method on unseen problems

General Probability Practice Workbook

30 original general probability problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.

See the cover and sample pages

Error diagnosis

Why the other letters tempt

These notes identify the calculation error associated with each wrong letter when that error is reproducible.

AThis is the next-trial success probability under the ordinary model alone and ignores the evidence favoring the always-successful model.
BThe value 9/16 is the unconditional probability of three consecutive successes, not the conditional probability of the third success after two have been observed.
CThe value 5/8 is the unconditional probability of the two observed successes. It is the denominator of the conditional calculation, not its result.
DThe value 4/5 is the posterior probability that the always-successful model was selected. A remaining 1/5 posterior weight still has next-trial success probability 1/2.

Original practice · fully worked

Original variant: prediction after an unrecorded color match

A container holds five silver tokens and three black tokens. Tokens are drawn one at a time without replacement. The first two drawn tokens have matching colors, but their color was not recorded. Calculate the probability that the third token drawn is silver.

A
3/13
B
1/2
C
15/26
D
5/8
E
10/13

Answer and calculation

Given a match, the first pair is silver with posterior probability 10/13 and black with probability 3/13. Averaging the remaining silver fractions gives 15/26, so choice C.

Setup

Let M be the event that the first two colors match. Count the possible unordered first pairs by color.

\Pr(SS)=\frac{\binom52}{\binom82}=\frac{10}{28}

\Pr(BB)=\frac{\binom32}{\binom82}=\frac3{28}

\Pr(M)=\frac{13}{28}

Model

Update the color of the matched pair, then use the corresponding composition of the six remaining tokens.

\Pr(SS\mid M)=\frac{10}{13},\qquad \Pr(BB\mid M)=\frac3{13}

\Pr(S_3\mid SS)=\frac36,\qquad \Pr(S_3\mid BB)=\frac56

Compute

Average the two possible remaining silver fractions using the posterior pair-color probabilities.

\Pr(S_3\mid M)=\frac{10}{13}\frac36+\frac3{13}\frac56

\Pr(S_3\mid M)=\frac{15}{26}=0.576923\ldots

Answer

The third token is silver with probability 15/26.

\boxed{\frac{15}{26}\quad\text{(C)}}

Keep practicing

Next steps for Exam P General Probability

Work through the related free solutions for this topic, review the formulas you used, or keep every current Exam P solution in one searchable PDF.

More Exam P General Probability practice problems Free Exam P formula sheet Exam P study manual PDF — $49

How to solve this Conditional Probability question

Setup

Model

Compute

Answer

General Probability Practice Workbook

Why the other letters tempt

Original variant: prediction after an unrecorded color match

Answer and calculation

Setup

Model

Compute

Answer

Related Exam P solutions

Next steps for Exam P General Probability