QUANTITATIVE PROFICIENCY TEST WORLDQUANT
UNIVERSITY 2026/2027 | Complete Solution | MSc
Admissions | QPT | Pass Guaranteed - A+ Graded
Section 1: Probability Theory - Distributions, Expectation & Bayes
(Questions 1-15)
Question 1
A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls is
7?
A. 1/12
B. 1/9
C. 1/6 [CORRECT]
D. 1/4
Rationale: There are 36 equally likely outcomes when rolling two dice. The pairs that
sum to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — 6 outcomes. Thus P(sum=7) = 6/36 =
1/6. A is too low (only counts some pairs), B is incorrect arithmetic, D overcounts.
Correct Answer: C
Question 2
,A random variable X follows a binomial distribution with n = 10 and p = 0.3. What is
E[X]?
A. 0.3
B. 3 [CORRECT]
C. 7
D. 10
Rationale: For a binomial distribution, E[X] = np = 10 × 0.3 = 3. A is just p, C is n(1-p), D is
n.
Correct Answer: B
Question 3
In a population, 1% of people have a certain disease. A test for the disease is 99%
accurate (both sensitivity and specificity are 99%). If a randomly selected person tests
positive, what is the probability they actually have the disease?
A. 0.99
B. 0.50 [CORRECT]
C. 0.10
D. 0.01
Rationale: Using Bayes' theorem: P(Disease|Positive) = (0.01 × 0.99) / [(0.01 × 0.99) +
(0.99 × 0.01)] = 0..0198 = 0.50. A ignores base rate, C and D are incorrect
applications.
Correct Answer: B
Question 4
,Let X ~ N(μ = 5, σ² = 4). What is P(X > 7)?
A. 0.1587 [CORRECT]
B. 0.0228
C. 0.3413
D. 0.5000
Rationale: Standardize: Z = (7-5)/2 = 1. P(X > 7) = P(Z > 1) = 1 - Φ(1) ≈ 0.1587. B is P(Z >
2), C is P(0 < Z < 1), D is the median probability.
Correct Answer: A
Question 5
Events A and B are independent with P(A) = 0.4 and P(B) = 0.5. What is P(A ∪ B)?
A. 0.20
B. 0.70 [CORRECT]
C. 0.90
D. 0.40
Rationale: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.4 + 0.5 - (0.4 × 0.5) = 0.9 - 0.2 = 0.70. A
is P(A ∩ B), C forgets to subtract intersection, D is just P(A).
Correct Answer: B
Question 6
A Poisson random variable has parameter λ = 4. What is the variance of this random
variable?
A. 2
, B. 4 [CORRECT]
C. 16
D. 8
Rationale: For a Poisson distribution, both mean and variance equal λ. Thus Var(X) = 4.
A is √λ, C is λ², D is 2λ.
Correct Answer: B
Question 7
A continuous random variable X has PDF f(x) = 2x for 0 ≤ x ≤ 1, and 0 otherwise. What is
E[X]?
A. 1/3
B. 2/3 [CORRECT]
C. 1/2
D. 1
Rationale: E[X] = ∫₀¹ x · 2x dx = ∫₀¹ 2x² dx = [2x³/3]₀¹ = 2/3. A is the median, C assumes
uniform, D is the upper bound.
Correct Answer: B
Question 8
The lifetime of a component follows an exponential distribution with mean 1000 hours.
What is the probability that the component lasts more than 2000 hours?
A. e⁻¹
B. e⁻² [CORRECT]
UNIVERSITY 2026/2027 | Complete Solution | MSc
Admissions | QPT | Pass Guaranteed - A+ Graded
Section 1: Probability Theory - Distributions, Expectation & Bayes
(Questions 1-15)
Question 1
A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls is
7?
A. 1/12
B. 1/9
C. 1/6 [CORRECT]
D. 1/4
Rationale: There are 36 equally likely outcomes when rolling two dice. The pairs that
sum to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — 6 outcomes. Thus P(sum=7) = 6/36 =
1/6. A is too low (only counts some pairs), B is incorrect arithmetic, D overcounts.
Correct Answer: C
Question 2
,A random variable X follows a binomial distribution with n = 10 and p = 0.3. What is
E[X]?
A. 0.3
B. 3 [CORRECT]
C. 7
D. 10
Rationale: For a binomial distribution, E[X] = np = 10 × 0.3 = 3. A is just p, C is n(1-p), D is
n.
Correct Answer: B
Question 3
In a population, 1% of people have a certain disease. A test for the disease is 99%
accurate (both sensitivity and specificity are 99%). If a randomly selected person tests
positive, what is the probability they actually have the disease?
A. 0.99
B. 0.50 [CORRECT]
C. 0.10
D. 0.01
Rationale: Using Bayes' theorem: P(Disease|Positive) = (0.01 × 0.99) / [(0.01 × 0.99) +
(0.99 × 0.01)] = 0..0198 = 0.50. A ignores base rate, C and D are incorrect
applications.
Correct Answer: B
Question 4
,Let X ~ N(μ = 5, σ² = 4). What is P(X > 7)?
A. 0.1587 [CORRECT]
B. 0.0228
C. 0.3413
D. 0.5000
Rationale: Standardize: Z = (7-5)/2 = 1. P(X > 7) = P(Z > 1) = 1 - Φ(1) ≈ 0.1587. B is P(Z >
2), C is P(0 < Z < 1), D is the median probability.
Correct Answer: A
Question 5
Events A and B are independent with P(A) = 0.4 and P(B) = 0.5. What is P(A ∪ B)?
A. 0.20
B. 0.70 [CORRECT]
C. 0.90
D. 0.40
Rationale: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.4 + 0.5 - (0.4 × 0.5) = 0.9 - 0.2 = 0.70. A
is P(A ∩ B), C forgets to subtract intersection, D is just P(A).
Correct Answer: B
Question 6
A Poisson random variable has parameter λ = 4. What is the variance of this random
variable?
A. 2
, B. 4 [CORRECT]
C. 16
D. 8
Rationale: For a Poisson distribution, both mean and variance equal λ. Thus Var(X) = 4.
A is √λ, C is λ², D is 2λ.
Correct Answer: B
Question 7
A continuous random variable X has PDF f(x) = 2x for 0 ≤ x ≤ 1, and 0 otherwise. What is
E[X]?
A. 1/3
B. 2/3 [CORRECT]
C. 1/2
D. 1
Rationale: E[X] = ∫₀¹ x · 2x dx = ∫₀¹ 2x² dx = [2x³/3]₀¹ = 2/3. A is the median, C assumes
uniform, D is the upper bound.
Correct Answer: B
Question 8
The lifetime of a component follows an exponential distribution with mean 1000 hours.
What is the probability that the component lasts more than 2000 hours?
A. e⁻¹
B. e⁻² [CORRECT]
