AP Stats Probability and Distributions 2 — Questions and Answers
Question 1: A bag contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. What is the probability both are red?
- 2/15 (Correct answer)
- 4/25
- 1/6
- 8/45
Correct answer: 2/15
P(both red) = (4/10) × (3/9) = 12/90 = 2/15.
Question 2: A continuous random variable X has a uniform distribution on [2, 8]. What is P(3 ≤ X ≤ 6)?
- 0.50 (Correct answer)
- 0.40
- 0.60
- 0.75
Correct answer: 0.50
Length of interval is 6 − 2 = 4; P(3 ≤ X ≤ 6) = (6 − 3)/6 = 3/6 = 0.50.
Question 3: If P(A) = 0.5, P(B) = 0.4, and P(A ∩ B) = 0.2, are events A and B independent?
- Yes, because P(A)·P(B) = P(A ∩ B) (Correct answer)
- No, because P(A ∩ B) < P(A)
- No, because P(A) ≠ P(B)
- Yes, because P(A) + P(B) = 0.9
Correct answer: Yes, because P(A)·P(B) = P(A ∩ B)
A and B are independent when P(A)·P(B) = P(A ∩ B); here 0.5 × 0.4 = 0.20 ✓.
Question 4: For a geometric distribution with success probability p = 0.3, what is the expected number of trials until the first success?
- 3.33 (Correct answer)
- 0.30
- 2.10
- 7.00
Correct answer: 3.33
The mean of a geometric distribution is 1/p = 1/0.3 ≈ 3.33.
Question 5: Which of the following is a property of a valid probability distribution for a discrete random variable X?
- All probabilities are between 0 and 1 and sum to 1 (Correct answer)
- All probabilities must equal each other
- The sum of probabilities must equal the number of outcomes
- Probabilities can be negative if they sum to 1
Correct answer: All probabilities are between 0 and 1 and sum to 1
A valid discrete probability distribution requires 0 ≤ P(X = x) ≤ 1 for all x and ΣP(X = x) = 1.
Question 6: A normal distribution has mean 50 and standard deviation 5. What z-score corresponds to x = 42?
- −1.6 (Correct answer)
- 1.6
- −0.8
- 8.0
Correct answer: −1.6
z = (x − μ)/σ = (42 − 50)/5 = −8/5 = −1.6.
Question 7: Events A and B are mutually exclusive. If P(A) = 0.25 and P(B) = 0.35, what is P(A ∪ B)?
- 0.60 (Correct answer)
- 0.0875
- 0.40
- 1.00
Correct answer: 0.60
For mutually exclusive events, P(A ∪ B) = P(A) + P(B) = 0.25 + 0.35 = 0.60.
AP Stats Probability and Distributions 2
A bag contains 4 red and 6 blue marbles.
Two marbles are drawn without replacement.
What is the probability both are red?