Probability and Distributions Flashcards

A bag contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. What is the probability both are red?

Answer: 2/15

P(both red) = (4/10) × (3/9) = 12/90 = 2/15.

A continuous random variable X has a uniform distribution on [2, 8]. What is P(3 ≤ X ≤ 6)?

Answer: 0.50

Length of interval is 6 − 2 = 4; P(3 ≤ X ≤ 6) = (6 − 3)/6 = 3/6 = 0.50.

If P(A) = 0.5, P(B) = 0.4, and P(A ∩ B) = 0.2, are events A and B independent?

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 ✓.

For a geometric distribution with success probability p = 0.3, what is the expected number of trials until the first success?

Answer: 3.33

The mean of a geometric distribution is 1/p = 1/0.3 ≈ 3.33.

Which of the following is a property of a valid probability distribution for a discrete random variable X?

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.

A normal distribution has mean 50 and standard deviation 5. What z-score corresponds to x = 42?

Answer: −1.6

z = (x − μ)/σ = (42 − 50)/5 = −8/5 = −1.6.

Events A and B are mutually exclusive. If P(A) = 0.25 and P(B) = 0.35, what is P(A ∪ B)?

Answer: 0.60

For mutually exclusive events, P(A ∪ B) = P(A) + P(B) = 0.25 + 0.35 = 0.60.