STAT 410 EXAM 1 VERSION C QUESTIONS AND CORRECT
ANSWERS (VERIFIED ANSWERS) PLUS RATIONALES
SPRING 2026 Q&A | INSTANT DOWNLOAD PDF
Core Domains
Probability Axioms and Set Theory
Combinatorial Probability
Conditional Probability and Bayes' Theorem
Discrete Random Variables and Distributions
Continuous Random Variables and Distributions
Expectation, Variance, and Moments
Joint and Marginal Distributions
Independence and Covariance
Transformations of Random Variables
Limit Theorems and Inequalities
Introduction
This comprehensive examination assesses the foundational
probability and statistical principles required of
,undergraduate students at the University of Illinois Urbana-
Champaign. It evaluates set theory, combinatorial probability,
conditional probability, discrete and continuous random
variables, expectation, variance, joint distributions, and
transformations. The examination employs multiple-choice
and scenario-based questions aligned with the STAT 410
course objectives. Emphasis is placed on critical thinking,
mathematical reasoning, and the application of probabilistic
principles to real-world problems. This assessment prepares
candidates for the rigor of the STAT 410 Exam 1 Version C and
advanced study in statistics and probability.
SECTION ONE: QUESTIONS 1–100
1. A fair six-sided die is rolled twice. What is the probability
that the sum of the two rolls is exactly 7?
A. 1/12
B. 1/6
C. 5/36
D. 1/9
🟢 B. 1/6
🔴 RATIONALE: There are 36 equally likely outcomes when
rolling two dice. The outcomes that sum to 7 are (1,6), (2,5),
,(3,4), (4,3), (5,2), and (6,1)—a total of 6 outcomes. Therefore,
P(sum = 7) = 6/36 = 1/6.
2. Let A and B be events with P(A) = 0.4, P(B) = 0.5, and
P(A ∩ B) = 0.2. What is P(A ∪ B)?
A. 0.7
B. 0.9
C. 0.6
D. 0.8
🟢 A. 0.7
🔴 RATIONALE: By the inclusion-exclusion principle, P(A ∪
B) = P(A) + P(B) − P(A ∩ B) = 0.4 + 0.5 − 0.2 = 0.7.
3. A box contains 5 red balls and 3 blue balls. Two balls are
drawn without replacement. What is the probability that
both balls are red?
A. 5/28
B. 10/28
C. 25/64
D. 5/14
, 🟢 D. 5/14
🔴 RATIONALE: P(both red) = (5/8) × (4/7) = 20/56 = 5/14.
This uses the multiplication rule for dependent events.
4. If P(A | B) = 0.3, P(B) = 0.5, and P(A | B') = 0.6, what is
P(A)?
A. 0.45
B. 0.35
C. 0.50
D. 0.55
🟢 A. 0.45
🔴 RATIONALE: By the law of total probability, P(A) = P(A |
B)P(B) + P(A | B')P(B') = 0.3 × 0.5 + 0.6 × 0.5 = 0.15 + 0.30 =
0.45.
5. A random variable X has the following probability mass
function: P(X = 1) = 0.2, P(X = 2) = 0.5, P(X = 3) = 0.3.
What is E[X]?
A. 2.0
B. 2.1
ANSWERS (VERIFIED ANSWERS) PLUS RATIONALES
SPRING 2026 Q&A | INSTANT DOWNLOAD PDF
Core Domains
Probability Axioms and Set Theory
Combinatorial Probability
Conditional Probability and Bayes' Theorem
Discrete Random Variables and Distributions
Continuous Random Variables and Distributions
Expectation, Variance, and Moments
Joint and Marginal Distributions
Independence and Covariance
Transformations of Random Variables
Limit Theorems and Inequalities
Introduction
This comprehensive examination assesses the foundational
probability and statistical principles required of
,undergraduate students at the University of Illinois Urbana-
Champaign. It evaluates set theory, combinatorial probability,
conditional probability, discrete and continuous random
variables, expectation, variance, joint distributions, and
transformations. The examination employs multiple-choice
and scenario-based questions aligned with the STAT 410
course objectives. Emphasis is placed on critical thinking,
mathematical reasoning, and the application of probabilistic
principles to real-world problems. This assessment prepares
candidates for the rigor of the STAT 410 Exam 1 Version C and
advanced study in statistics and probability.
SECTION ONE: QUESTIONS 1–100
1. A fair six-sided die is rolled twice. What is the probability
that the sum of the two rolls is exactly 7?
A. 1/12
B. 1/6
C. 5/36
D. 1/9
🟢 B. 1/6
🔴 RATIONALE: There are 36 equally likely outcomes when
rolling two dice. The outcomes that sum to 7 are (1,6), (2,5),
,(3,4), (4,3), (5,2), and (6,1)—a total of 6 outcomes. Therefore,
P(sum = 7) = 6/36 = 1/6.
2. Let A and B be events with P(A) = 0.4, P(B) = 0.5, and
P(A ∩ B) = 0.2. What is P(A ∪ B)?
A. 0.7
B. 0.9
C. 0.6
D. 0.8
🟢 A. 0.7
🔴 RATIONALE: By the inclusion-exclusion principle, P(A ∪
B) = P(A) + P(B) − P(A ∩ B) = 0.4 + 0.5 − 0.2 = 0.7.
3. A box contains 5 red balls and 3 blue balls. Two balls are
drawn without replacement. What is the probability that
both balls are red?
A. 5/28
B. 10/28
C. 25/64
D. 5/14
, 🟢 D. 5/14
🔴 RATIONALE: P(both red) = (5/8) × (4/7) = 20/56 = 5/14.
This uses the multiplication rule for dependent events.
4. If P(A | B) = 0.3, P(B) = 0.5, and P(A | B') = 0.6, what is
P(A)?
A. 0.45
B. 0.35
C. 0.50
D. 0.55
🟢 A. 0.45
🔴 RATIONALE: By the law of total probability, P(A) = P(A |
B)P(B) + P(A | B')P(B') = 0.3 × 0.5 + 0.6 × 0.5 = 0.15 + 0.30 =
0.45.
5. A random variable X has the following probability mass
function: P(X = 1) = 0.2, P(X = 2) = 0.5, P(X = 3) = 0.3.
What is E[X]?
A. 2.0
B. 2.1
