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Probability · foundations diagnostic PDF

Exam P General Probability: 30 Fully Worked Problems

Strengthen counting, event algebra, conditional probability, Bayes, total probability, and independence before distribution calculations take over.

One-time USD 19 purchase for a personalized PDF. The 30 original variants are also included in the complete Probability Proof Manual; choose this focused diagnostic for curated practice and a lower entry price.

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30
Pages
77
Time limit
180 min
Price
$19

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Cover of General Probability Practice Workbook
Version 1.0.0 · Updated 2026-08-24

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One original Exam P practice question with full solution

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.

  1. 0.160
  2. 0.200
  3. 0.300
  4. 0.500
  5. 0.800

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

Compute the ordered success-then-failure likelihood under each process mode.

LA=0.80(0.20)=0.16L_A=0.80(0.20)=0.16
LB=0.20(0.80)=0.16L_B=0.20(0.80)=0.16

Model

Apply Bayes' formula by multiplying each equal likelihood by its mode prior and normalizing the mode-A weight.

Pr⁡(A∣SF)=0.30LA0.30LA+0.70LB\Pr(A\mid SF)=\frac{0.30L_A}{0.30L_A+0.70L_B}

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\Pr(A\mid SF)=0.048/(0.048+0.112)=0.30

Answer

The posterior probability of mode A is 0.300, which is choice C.

0.30(C)\boxed{0.30\quad\text{(C)}}

Why the other choices are wrong

A
The value 0.160 is the likelihood of the ordered mixed outcome under either mode, not a posterior mode probability.
B
The value 0.200 is the single-trial success probability in mode B and does not incorporate the mode prior or both observations.
D
The value 0.500 results from normalizing the two equal likelihoods without their unequal prior weights.
E
The value 0.800 is the single-trial success probability in mode A, not the posterior probability of mode A.

Independent answer check

Primary value0.3
Primary methodUse ordered two-trial likelihoods in Bayes' formula.
Independent value0.3
Independent methodThe likelihood ratio is one, so posterior odds equal prior odds 3:7.
AnswerC · agreement passed

Contents

Inside the 77-page edition

Candidates preparing in English for the Probability (P) Exam

Table of contents

  1. 30 original general probability problems
  2. Counting and event algebra
  3. Conditional probability and Bayes
  4. Independence and total probability
  5. 120 wrong-choice analyses and two-path checks

What is included

30 verified original variants selected without duplication for this focused diagnostic. Official problem wording is not included.

Verification

Every included result is reproducible

Every selected problem already passed a reproducible calculation, an independent second method, and answer-letter agreement. The compiler rejects any failed check, cross-product duplicate, or official source wording.

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Questions about this edition

Are these official sample questions?

No. Every question is an independently written original variant. Official problem wording is not reproduced.

Are complete solutions included?

Yes. Every problem includes Setup, Model, Compute, and Answer steps plus a concrete explanation for each of the four incorrect choices.

How are the answers checked?

Each answer must agree under a reproducible primary calculation and a distinct independent method before the PDF can build.

Does this overlap the complete proof manual?

Yes. These 30 original variants are selected from the verified variant bank included in the Probability Proof Manual. This edition adds a curated focused diagnostic, answer sheet, answer key, and focused purchase price.

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General Probability Practice Workbook

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