SOPHIA STATISTICS UNIT 4 MILESTONE 2026/2027 |
Questions Answers & Rationale | Summer Updated | Pass
Guaranteed - A+ Graded
[SECTION 1: PROBABILITY CONCEPTS & RULES (Q1-15)]
Q1. A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If one marble
is selected at random, what is the probability of selecting a blue marble?
A. 3/10 [CORRECT]
B. 1/3
C. 3/5
D. 2/5
Correct Answer: A. 3/10 [CORRECT]
Rationale: Classical probability: P(blue) = number of favorable outcomes / total
outcomes = 3/(5+3+2) = 3/10. Option B incorrectly uses only blue and green. Option
C uses red and blue. Option D uses green and blue. Statistical Principle: Classical
probability assumes equally likely outcomes. Always verify the total sample space.
Q2. Events A and B are mutually exclusive. P(A) = 0.4 and P(B) = 0.3. What is P(A or
B)?
A. 0.12
B. 0.7 [CORRECT]
C. 1.0
D. 0.1
Correct Answer: B. 0.7 [CORRECT]
,Rationale: For mutually exclusive (disjoint) events, P(A or B) = P(A) + P(B) = 0.4 + 0.3
= 0.7. No subtraction needed since P(A and B) = 0. Option A uses multiplication
(independence rule). Option C exceeds 1. Option D subtracts instead of adds.
Statistical Principle: Addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). For mutually
exclusive events, P(A ∩ B) = 0.
Q3. If P(A) = 0.6, P(B) = 0.5, and P(A and B) = 0.2, what is P(A or B)?
A. 1.1
B. 0.9 [CORRECT]
C. 0.3
D. 0.8
Correct Answer: B. 0.9 [CORRECT]
Rationale: General addition rule: P(A or B) = P(A) + P(B) - P(A and B) = 0.6 + 0.5 - 0.2
= 0.9. Option A forgets to subtract the intersection. Option C multiplies (wrong rule).
Option D uses incorrect arithmetic. Statistical Principle: Always subtract P(A ∩ B) to
avoid double-counting outcomes in both events.
Q4. A diagnostic test has sensitivity 0.95 and specificity 0.90. The disease prevalence
is 0.02. Using Bayes' theorem, what is the probability of having the disease given a
positive test result?
A. 0.95
B. 0.02
C. Approximately 0.162 [CORRECT]
D. 0.90
Correct Answer: C. Approximately 0.162 [CORRECT]
Rationale: Bayes' theorem: P(D|T⁺) = [P(T⁺|D) × P(D)] / [P(T⁺|D) × P(D) + P(T⁺|ND) ×
P(ND)] = (0.95 × 0.02) / [(0.95 × 0.02) + (0.10 × 0.98)] = 0.019 / (0.019 + 0.098) =
0.019/0.117 ≈ 0.162. Option A is sensitivity. Option B is prevalence. Option D is
,specificity. Statistical Principle: Even with high sensitivity/specificity, low prevalence
yields many false positives. This is why screening low-prevalence populations
requires confirmatory testing.
Q5. If P(A) = 0.3 and P(B|A) = 0.4, what is P(A and B)?
A. 0.7
B. 1.2
C. 0.12 [CORRECT]
D. 0.1
Correct Answer: C. 0.12 [CORRECT]
Rationale: Multiplication rule: P(A and B) = P(A) × P(B|A) = 0.3 × 0.4 = 0.12. Option A
adds. Option B exceeds 1 (impossible). Option D is incorrect division. Statistical
Principle: For dependent events, use conditional probability. For independent
events, P(B|A) = P(B), so P(A and B) = P(A) × P(B).
Q6. Events A and B are independent. P(A) = 0.4 and P(B) = 0.5. What is P(A and B)?
A. 0.9
B. 0.2 [CORRECT]
C. 0.1
D. 0.45
Correct Answer: B. 0.2 [CORRECT]
Rationale: For independent events: P(A and B) = P(A) × P(B) = 0.4 × 0.5 = 0.2.
Option A adds (addition rule). Option C is incorrect. Option D uses wrong
multiplication. Statistical Principle: Independence means knowing A occurred
doesn't change P(B). Verify independence from problem context—don't assume it.
, Q7. The complement of event A has probability 0.7. What is P(A)?
A. 0.7
B. 1.0
C. 0.3 [CORRECT]
D. 0.0
Correct Answer: C. 0.3 [CORRECT]
Rationale: Complementary events: P(A) = 1 - P(A') = 1 - 0.7 = 0.3. Option A is P(A').
Option B is the total probability. Option D is incorrect subtraction. Statistical
Principle: P(A) + P(A') = 1. Complementary probability is useful when P(A) is difficult
to calculate directly but P(A') is easier.
Q8. A card is drawn from a standard deck. What is the probability of drawing a heart
or a queen?
A. 16/52
B. 17/52
C. 4/13 [CORRECT]
D. 1/4
Correct Answer: C. 4/13 [CORRECT]
Rationale: P(heart or queen) = P(heart) + P(queen) - P(heart and queen) = 13/52 +
4/52 - 1/52 = 16/52 = 4/13. Option A forgets to subtract the queen of hearts. Option
B adds incorrectly. Option D is P(heart) only. Statistical Principle: "Or" problems
usually require subtraction of the intersection unless events are mutually exclusive.
Q9. A fair die is rolled twice. What is the probability of getting a sum of 7?
A. 1/6 [CORRECT]
B. 1/12
C. 1/3
D. 1/2
Questions Answers & Rationale | Summer Updated | Pass
Guaranteed - A+ Graded
[SECTION 1: PROBABILITY CONCEPTS & RULES (Q1-15)]
Q1. A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If one marble
is selected at random, what is the probability of selecting a blue marble?
A. 3/10 [CORRECT]
B. 1/3
C. 3/5
D. 2/5
Correct Answer: A. 3/10 [CORRECT]
Rationale: Classical probability: P(blue) = number of favorable outcomes / total
outcomes = 3/(5+3+2) = 3/10. Option B incorrectly uses only blue and green. Option
C uses red and blue. Option D uses green and blue. Statistical Principle: Classical
probability assumes equally likely outcomes. Always verify the total sample space.
Q2. Events A and B are mutually exclusive. P(A) = 0.4 and P(B) = 0.3. What is P(A or
B)?
A. 0.12
B. 0.7 [CORRECT]
C. 1.0
D. 0.1
Correct Answer: B. 0.7 [CORRECT]
,Rationale: For mutually exclusive (disjoint) events, P(A or B) = P(A) + P(B) = 0.4 + 0.3
= 0.7. No subtraction needed since P(A and B) = 0. Option A uses multiplication
(independence rule). Option C exceeds 1. Option D subtracts instead of adds.
Statistical Principle: Addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). For mutually
exclusive events, P(A ∩ B) = 0.
Q3. If P(A) = 0.6, P(B) = 0.5, and P(A and B) = 0.2, what is P(A or B)?
A. 1.1
B. 0.9 [CORRECT]
C. 0.3
D. 0.8
Correct Answer: B. 0.9 [CORRECT]
Rationale: General addition rule: P(A or B) = P(A) + P(B) - P(A and B) = 0.6 + 0.5 - 0.2
= 0.9. Option A forgets to subtract the intersection. Option C multiplies (wrong rule).
Option D uses incorrect arithmetic. Statistical Principle: Always subtract P(A ∩ B) to
avoid double-counting outcomes in both events.
Q4. A diagnostic test has sensitivity 0.95 and specificity 0.90. The disease prevalence
is 0.02. Using Bayes' theorem, what is the probability of having the disease given a
positive test result?
A. 0.95
B. 0.02
C. Approximately 0.162 [CORRECT]
D. 0.90
Correct Answer: C. Approximately 0.162 [CORRECT]
Rationale: Bayes' theorem: P(D|T⁺) = [P(T⁺|D) × P(D)] / [P(T⁺|D) × P(D) + P(T⁺|ND) ×
P(ND)] = (0.95 × 0.02) / [(0.95 × 0.02) + (0.10 × 0.98)] = 0.019 / (0.019 + 0.098) =
0.019/0.117 ≈ 0.162. Option A is sensitivity. Option B is prevalence. Option D is
,specificity. Statistical Principle: Even with high sensitivity/specificity, low prevalence
yields many false positives. This is why screening low-prevalence populations
requires confirmatory testing.
Q5. If P(A) = 0.3 and P(B|A) = 0.4, what is P(A and B)?
A. 0.7
B. 1.2
C. 0.12 [CORRECT]
D. 0.1
Correct Answer: C. 0.12 [CORRECT]
Rationale: Multiplication rule: P(A and B) = P(A) × P(B|A) = 0.3 × 0.4 = 0.12. Option A
adds. Option B exceeds 1 (impossible). Option D is incorrect division. Statistical
Principle: For dependent events, use conditional probability. For independent
events, P(B|A) = P(B), so P(A and B) = P(A) × P(B).
Q6. Events A and B are independent. P(A) = 0.4 and P(B) = 0.5. What is P(A and B)?
A. 0.9
B. 0.2 [CORRECT]
C. 0.1
D. 0.45
Correct Answer: B. 0.2 [CORRECT]
Rationale: For independent events: P(A and B) = P(A) × P(B) = 0.4 × 0.5 = 0.2.
Option A adds (addition rule). Option C is incorrect. Option D uses wrong
multiplication. Statistical Principle: Independence means knowing A occurred
doesn't change P(B). Verify independence from problem context—don't assume it.
, Q7. The complement of event A has probability 0.7. What is P(A)?
A. 0.7
B. 1.0
C. 0.3 [CORRECT]
D. 0.0
Correct Answer: C. 0.3 [CORRECT]
Rationale: Complementary events: P(A) = 1 - P(A') = 1 - 0.7 = 0.3. Option A is P(A').
Option B is the total probability. Option D is incorrect subtraction. Statistical
Principle: P(A) + P(A') = 1. Complementary probability is useful when P(A) is difficult
to calculate directly but P(A') is easier.
Q8. A card is drawn from a standard deck. What is the probability of drawing a heart
or a queen?
A. 16/52
B. 17/52
C. 4/13 [CORRECT]
D. 1/4
Correct Answer: C. 4/13 [CORRECT]
Rationale: P(heart or queen) = P(heart) + P(queen) - P(heart and queen) = 13/52 +
4/52 - 1/52 = 16/52 = 4/13. Option A forgets to subtract the queen of hearts. Option
B adds incorrectly. Option D is P(heart) only. Statistical Principle: "Or" problems
usually require subtraction of the intersection unless events are mutually exclusive.
Q9. A fair die is rolled twice. What is the probability of getting a sum of 7?
A. 1/6 [CORRECT]
B. 1/12
C. 1/3
D. 1/2
