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INTRODUCTION TO PROBABILITY MODELS, 13TH EDITION
(SHELDON M. ROSS) EXAM 2026 QUESTIONS WITH VERIFIED
QUESTIONS DETAILED RATIONALES GRADED A+
INTRODUCTION TO PROBABILITY MODELS, 13th Edition (Sheldon M. Ross) — EXAM
10-POINT EXAM COVERAGE SUMMARY
Introduction to Probability Theory – Sample spaces, axioms of probability, combinatorics,
conditional probability, Bayes' formula, independence
Random Variables – Discrete and continuous random variables, expectation, variance,
Bernoulli, binomial, Poisson, uniform, normal, exponential distributions
Conditional Probability and Conditional Expectation – Conditional distributions, conditional
expectation, computing expectations by conditioning, martingales
Markov Chains – Chapman-Kolmogorov equations, classification of states, limiting probabilities,
transient analysis, branching processes
The Exponential Distribution and the Poisson Process – Memoryless property, Poisson process
axioms, interarrival times, nonhomogeneous Poisson process, compound Poisson process
Continuous-Time Markov Chains – Birth and death processes, transition probabilities, limiting
probabilities, time reversibility
Renewal Theory and Its Applications – Renewal function, renewal reward processes,
regenerative processes, inspection paradox
Queueing Theory – M/M/1, M/M/s, M/G/1 queues, Little's formula, priority queues, queueing
networks
Reliability Theory – Structure functions, system reliability, hazard rate functions, exponential
and Weibull lifetimes
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Brownian Motion, Simulation, Coupling & Martingales – Standard Brownian motion, hitting
times, simulation techniques, coupling, martingale properties
SECTION 1: INTRODUCTION TO PROBABILITY THEORY (CHAPTER 1) (25 QUESTIONS)
1. A quality control engineer tests a batch of 100 microchips and finds that 8 are defective. If
one chip is selected at random, the probability it is not defective is which of the following?
A) 0.08
B) 0.92
C) 0.85
D) 0.95
Rationale: The probability of a defective chip is 8/100 = 0.08, so the probability of not being
defective is 1 − 0.08 = 0.92.
2. In a sample space S with events A and B, if P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2, what is
P(A ∪ B)?
A) 0.70
B) 0.90
C) 0.20
D) 0.10
Rationale: By the addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.4 + 0.5 − 0.2 = 0.70.
3. If two fair coins are flipped, what is the probability of getting at least one heads?
A) 1/4
B) 1/2
C) 3/4
D) 1/3
Rationale: P(at least one heads) = 1 − P(no heads) = 1 − (1/2 × 1/2) = 3/4.
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4. A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?
A) 1/6
B) 1/3
C) 1/2
D) 2/3
Rationale: Numbers greater than 4 are 5 and 6. P = 2/6 = 1/3.
5. Two events A and B are independent. If P(A) = 0.3 and P(B) = 0.4, what is P(A ∩ B)?
A) 0.12
B) 0.70
C) 0.10
D) 0.30
Rationale: For independent events, P(A ∩ B) = P(A) × P(B) = 0.3 × 0.4 = 0.12.
6. A bag contains 5 red marbles and 3 blue marbles. If one marble is drawn at random, what is
the probability it is red?
A) 3/8
B) 5/8
C) 5/3
D) 1/2
Rationale: P(red) = 5/(5+3) = 5/8.
7. A card is drawn from a standard 52-card deck. What is the probability of drawing a heart?
A) 1/4
B) 1/13
C) 1/52
D) 1/2
Rationale: There are 13 hearts in a 52-card deck. P = 13/52 = 1/4.
8. If P(A) = 0.6, what is P(A^c)?
A) 0.6
B) 0.4
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C) 1.6
D) 0.3
Rationale: The complement rule: P(A^c) = 1 − P(A) = 1 − 0.6 = 0.4.
9. A die is rolled twice. What is the probability of rolling two sixes?
A) 1/36
B) 1/12
C) 1/6
D) 1/4
Rationale: P(6 and 6) = 1/6 × 1/6 = 1/36.
10. In a survey, 40% of respondents like coffee, 30% like tea, and 15% like both. What is the
probability a randomly selected respondent likes coffee or tea?
A) 0.55
B) 0.70
C) 0.45
D) 0.85
Rationale: P(C ∪ T) = 0.40 + 0.30 − 0.15 = 0.55.
11. If events A and B are mutually exclusive, what is P(A ∩ B)?
A) 0
B) 1
C) P(A) + P(B)
D) P(A) × P(B)
Rationale: Mutually exclusive events cannot occur simultaneously, so P(A ∩ B) = 0.
12. A jar contains 4 green, 6 yellow, and 5 orange candies. If one candy is selected at random,
what is the probability it is NOT green?
A) 4/15
B) 11/15
C) 6/15
INTRODUCTION TO PROBABILITY MODELS, 13TH EDITION
(SHELDON M. ROSS) EXAM 2026 QUESTIONS WITH VERIFIED
QUESTIONS DETAILED RATIONALES GRADED A+
INTRODUCTION TO PROBABILITY MODELS, 13th Edition (Sheldon M. Ross) — EXAM
10-POINT EXAM COVERAGE SUMMARY
Introduction to Probability Theory – Sample spaces, axioms of probability, combinatorics,
conditional probability, Bayes' formula, independence
Random Variables – Discrete and continuous random variables, expectation, variance,
Bernoulli, binomial, Poisson, uniform, normal, exponential distributions
Conditional Probability and Conditional Expectation – Conditional distributions, conditional
expectation, computing expectations by conditioning, martingales
Markov Chains – Chapman-Kolmogorov equations, classification of states, limiting probabilities,
transient analysis, branching processes
The Exponential Distribution and the Poisson Process – Memoryless property, Poisson process
axioms, interarrival times, nonhomogeneous Poisson process, compound Poisson process
Continuous-Time Markov Chains – Birth and death processes, transition probabilities, limiting
probabilities, time reversibility
Renewal Theory and Its Applications – Renewal function, renewal reward processes,
regenerative processes, inspection paradox
Queueing Theory – M/M/1, M/M/s, M/G/1 queues, Little's formula, priority queues, queueing
networks
Reliability Theory – Structure functions, system reliability, hazard rate functions, exponential
and Weibull lifetimes
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Brownian Motion, Simulation, Coupling & Martingales – Standard Brownian motion, hitting
times, simulation techniques, coupling, martingale properties
SECTION 1: INTRODUCTION TO PROBABILITY THEORY (CHAPTER 1) (25 QUESTIONS)
1. A quality control engineer tests a batch of 100 microchips and finds that 8 are defective. If
one chip is selected at random, the probability it is not defective is which of the following?
A) 0.08
B) 0.92
C) 0.85
D) 0.95
Rationale: The probability of a defective chip is 8/100 = 0.08, so the probability of not being
defective is 1 − 0.08 = 0.92.
2. In a sample space S with events A and B, if P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2, what is
P(A ∪ B)?
A) 0.70
B) 0.90
C) 0.20
D) 0.10
Rationale: By the addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.4 + 0.5 − 0.2 = 0.70.
3. If two fair coins are flipped, what is the probability of getting at least one heads?
A) 1/4
B) 1/2
C) 3/4
D) 1/3
Rationale: P(at least one heads) = 1 − P(no heads) = 1 − (1/2 × 1/2) = 3/4.
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4. A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?
A) 1/6
B) 1/3
C) 1/2
D) 2/3
Rationale: Numbers greater than 4 are 5 and 6. P = 2/6 = 1/3.
5. Two events A and B are independent. If P(A) = 0.3 and P(B) = 0.4, what is P(A ∩ B)?
A) 0.12
B) 0.70
C) 0.10
D) 0.30
Rationale: For independent events, P(A ∩ B) = P(A) × P(B) = 0.3 × 0.4 = 0.12.
6. A bag contains 5 red marbles and 3 blue marbles. If one marble is drawn at random, what is
the probability it is red?
A) 3/8
B) 5/8
C) 5/3
D) 1/2
Rationale: P(red) = 5/(5+3) = 5/8.
7. A card is drawn from a standard 52-card deck. What is the probability of drawing a heart?
A) 1/4
B) 1/13
C) 1/52
D) 1/2
Rationale: There are 13 hearts in a 52-card deck. P = 13/52 = 1/4.
8. If P(A) = 0.6, what is P(A^c)?
A) 0.6
B) 0.4
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C) 1.6
D) 0.3
Rationale: The complement rule: P(A^c) = 1 − P(A) = 1 − 0.6 = 0.4.
9. A die is rolled twice. What is the probability of rolling two sixes?
A) 1/36
B) 1/12
C) 1/6
D) 1/4
Rationale: P(6 and 6) = 1/6 × 1/6 = 1/36.
10. In a survey, 40% of respondents like coffee, 30% like tea, and 15% like both. What is the
probability a randomly selected respondent likes coffee or tea?
A) 0.55
B) 0.70
C) 0.45
D) 0.85
Rationale: P(C ∪ T) = 0.40 + 0.30 − 0.15 = 0.55.
11. If events A and B are mutually exclusive, what is P(A ∩ B)?
A) 0
B) 1
C) P(A) + P(B)
D) P(A) × P(B)
Rationale: Mutually exclusive events cannot occur simultaneously, so P(A ∩ B) = 0.
12. A jar contains 4 green, 6 yellow, and 5 orange candies. If one candy is selected at random,
what is the probability it is NOT green?
A) 4/15
B) 11/15
C) 6/15
