SOPHIA STATISTICS UNIT 3 MILESTONE COMPLETE |
Answers & Explanations | Pass Guaranteed - A+
Graded
Section 1: Probability Fundamentals (Q1-12)
Q1. Which of the following best describes empirical probability?
A. The probability assigned based on personal judgment or belief
B. The probability determined by the ratio of favorable outcomes to total possible
outcomes in a sample space
C. The probability estimated from the relative frequency of an event occurring over
many repetitions of an experiment
D. The probability calculated using combinatorial formulas for equally likely
outcomes
Correct Answer: C [CORRECT]
Rationale: Empirical probability is based on observed data and relative frequency
from repeated trials, not theoretical assumptions. Option A describes subjective
probability; Option B describes classical probability (theoretical); Option D also
describes classical probability using combinatorics. On Sophia exams, always
distinguish the source of the probability: classical = theory/equally likely, empirical =
data/frequency, subjective = opinion. Real-world tip: Weather forecasters use
empirical probability based on historical data patterns.
Q2. A quality control inspector randomly selects one item from a batch of 50 items,
where 8 are defective. What is the probability that the selected item is NOT
defective?
A. 0.16
B. 0.84
C. 0.80
D. 0.92
,Correct Answer: B [CORRECT]
Rationale: Using the complement rule: P(not defective) = 1 - P(defective) = 1 - (8/50)
= 1 - 0.16 = 0.84. Option A is P(defective), a common trap where students forget to
subtract from 1. Option C incorrectly subtracts 0.16 from 1.0 but miscalculates.
Option D incorrectly uses 46/50. Sophia trap: Always verify whether the question asks
for the event or its complement—read carefully for words like "not," "at least one," or
"none."
Q3. In a standard deck of 52 cards, what is the probability of drawing a heart OR a
king?
A. 17/52
B. 16/52
C. 4/13
D. 1/4
Correct Answer: C [CORRECT]
Rationale: Using the general addition rule: P(Heart ∪ King) = P(Heart) + P(King) -
P(Heart ∩ King) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13. Option A adds without
subtracting the intersection (the king of hearts), violating the addition rule for non-
mutually exclusive events. Option B is the correct count before simplification but not
simplified. Option D is just P(Heart). Sophia trap: Cards, dice, and marbles almost
always involve non-mutually exclusive events—never blindly add probabilities
without checking for overlap.
Q4. Events A and B are mutually exclusive. If P(A) = 0.35 and P(B) = 0.42, what is P(A
or B)?
A. 0.77
B. 0.147
C. 0.58
D. 0.07
, Correct Answer: A [CORRECT]
Rationale: For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B) =
0.35 + 0.42 = 0.77. Option B multiplies the probabilities (0.35 × 0.42), which is the
multiplication rule for independent events, not addition. Option C incorrectly
subtracts P(A) from 1. Option D subtracts the probabilities. Sophia trap: The phrase
"mutually exclusive" is your signal to use the simplified addition rule—if you see this,
do NOT subtract an intersection; there isn't one.
Q5. A bag contains 4 red, 5 blue, and 6 green marbles. Two marbles are drawn WITH
replacement. What is the probability that both are blue?
A. 25/225
B. 20/210
C. 1/9
D. 2/15
Correct Answer: C [CORRECT]
Rationale: With replacement, draws are independent: P(Blue₁ ∩ Blue₂) = P(Blue) ×
P(Blue) = (5/15) × (5/15) = 25/225 = 1/9. Option A gives the unsimplified fraction but
is mathematically equivalent; however, Sophia expects simplified answers. Option B
uses combinations for without replacement (5/15 × 4/14), a major trap. Option D
incorrectly adds probabilities. Sophia trap: "With replacement" = independent events,
multiply individual probabilities; "without replacement" = dependent events, adjust
the second probability. Underline this distinction in every problem.
Q6. Using the same bag (4 red, 5 blue, 6 green), two marbles are drawn WITHOUT
replacement. What is the probability the first is red AND the second is green?
A. 24/210
B. 4/35
C. 24/225
D. 8/75
Answers & Explanations | Pass Guaranteed - A+
Graded
Section 1: Probability Fundamentals (Q1-12)
Q1. Which of the following best describes empirical probability?
A. The probability assigned based on personal judgment or belief
B. The probability determined by the ratio of favorable outcomes to total possible
outcomes in a sample space
C. The probability estimated from the relative frequency of an event occurring over
many repetitions of an experiment
D. The probability calculated using combinatorial formulas for equally likely
outcomes
Correct Answer: C [CORRECT]
Rationale: Empirical probability is based on observed data and relative frequency
from repeated trials, not theoretical assumptions. Option A describes subjective
probability; Option B describes classical probability (theoretical); Option D also
describes classical probability using combinatorics. On Sophia exams, always
distinguish the source of the probability: classical = theory/equally likely, empirical =
data/frequency, subjective = opinion. Real-world tip: Weather forecasters use
empirical probability based on historical data patterns.
Q2. A quality control inspector randomly selects one item from a batch of 50 items,
where 8 are defective. What is the probability that the selected item is NOT
defective?
A. 0.16
B. 0.84
C. 0.80
D. 0.92
,Correct Answer: B [CORRECT]
Rationale: Using the complement rule: P(not defective) = 1 - P(defective) = 1 - (8/50)
= 1 - 0.16 = 0.84. Option A is P(defective), a common trap where students forget to
subtract from 1. Option C incorrectly subtracts 0.16 from 1.0 but miscalculates.
Option D incorrectly uses 46/50. Sophia trap: Always verify whether the question asks
for the event or its complement—read carefully for words like "not," "at least one," or
"none."
Q3. In a standard deck of 52 cards, what is the probability of drawing a heart OR a
king?
A. 17/52
B. 16/52
C. 4/13
D. 1/4
Correct Answer: C [CORRECT]
Rationale: Using the general addition rule: P(Heart ∪ King) = P(Heart) + P(King) -
P(Heart ∩ King) = 13/52 + 4/52 - 1/52 = 16/52 = 4/13. Option A adds without
subtracting the intersection (the king of hearts), violating the addition rule for non-
mutually exclusive events. Option B is the correct count before simplification but not
simplified. Option D is just P(Heart). Sophia trap: Cards, dice, and marbles almost
always involve non-mutually exclusive events—never blindly add probabilities
without checking for overlap.
Q4. Events A and B are mutually exclusive. If P(A) = 0.35 and P(B) = 0.42, what is P(A
or B)?
A. 0.77
B. 0.147
C. 0.58
D. 0.07
, Correct Answer: A [CORRECT]
Rationale: For mutually exclusive events, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B) =
0.35 + 0.42 = 0.77. Option B multiplies the probabilities (0.35 × 0.42), which is the
multiplication rule for independent events, not addition. Option C incorrectly
subtracts P(A) from 1. Option D subtracts the probabilities. Sophia trap: The phrase
"mutually exclusive" is your signal to use the simplified addition rule—if you see this,
do NOT subtract an intersection; there isn't one.
Q5. A bag contains 4 red, 5 blue, and 6 green marbles. Two marbles are drawn WITH
replacement. What is the probability that both are blue?
A. 25/225
B. 20/210
C. 1/9
D. 2/15
Correct Answer: C [CORRECT]
Rationale: With replacement, draws are independent: P(Blue₁ ∩ Blue₂) = P(Blue) ×
P(Blue) = (5/15) × (5/15) = 25/225 = 1/9. Option A gives the unsimplified fraction but
is mathematically equivalent; however, Sophia expects simplified answers. Option B
uses combinations for without replacement (5/15 × 4/14), a major trap. Option D
incorrectly adds probabilities. Sophia trap: "With replacement" = independent events,
multiply individual probabilities; "without replacement" = dependent events, adjust
the second probability. Underline this distinction in every problem.
Q6. Using the same bag (4 red, 5 blue, 6 green), two marbles are drawn WITHOUT
replacement. What is the probability the first is red AND the second is green?
A. 24/210
B. 4/35
C. 24/225
D. 8/75
