Q: A bag contains 7 green and 5 yellow marbles. Two are drawn without replacement. What is the probability both are green?

$$\frac{7}{12} \times \frac{6}{11}$$

Q: For events A and B, $$P(A) = 0.4$$, $$P(B) = 0.5$$, $$P(A \text{ and } B) = 0.2. $$Are A and B independent?

Yes, because $$P(A) \times P(B) = P(A \text{ and } B)$$

Q: A game costs $$3 to $$play. You win $$10$$ with probability $$0.2$$ and $$0$$ otherwise. What is the expected net profit per play?

$$0.2 \times 7 + 0.8 \times (-3) = -1$$

Q: For $$X \sim \text{Binomial}(n=8, p=0.25)$$, what is the probability of exactly 2 successes?

$$C(8,2) \times (0.25)^2$ \times (0.75)^6$

Q: For $$X \sim \text{Binomial}(n=100, p=0.6)$$, what is the standard deviation?

$$\sqrt{100 \times 0.6 \times 0.4}$$

Question 1

A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?

Accepted Answer

$$\frac{1}{3}$$

Question 2

In a deck of 52 cards, what is the probability of drawing a queen or a club?

Accepted Answer

$$\frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$$

Question 3

Which of the following best describes the difference between independent and disjoint events?

Accepted Answer

Disjoint events cannot occur together, while independent events do not affect each other's probability.

Question 4

A device works on any day with probability 0.98, independently across 6 days. What is the probability that it fails at least once during those 6 days?

Accepted Answer

$$1 - (0.98)^6$

Question 5

A bag contains 7 green and 5 yellow marbles. Two are drawn without replacement. What is the probability both are green?

Accepted Answer

$$\frac{7}{12} 	imes \frac{6}{11}$$

Question 6

For events A and B, $$P(A) = 0.4$$, $$P(B) = 0.5$$, $$P(A 	ext{ and } B) = 0.2. $$Are A and B independent?

Accepted Answer

Yes, because $$P(A) 	imes P(B) = P(A 	ext{ and } B)$$

Question 7

Which of the following is NOT a requirement for a valid discrete probability distribution?

Accepted Answer

Probabilities can be negative if the sum is 1.

Question 8

A game costs $$3 to $$play. You win $$10$$ with probability $$0.2$$ and $$0$$ otherwise. What is the expected net profit per play?

Accepted Answer

$$0.2 	imes 7 + 0.8 	imes (-3) = -1$$

Question 9

For $$X \sim 	ext{Binomial}(n=8, p=0.25)$$, what is the probability of exactly 2 successes?

Accepted Answer

$$C(8,2) 	imes (0.25)^2$ 	imes (0.75)^6$

Question 10

For $$X \sim 	ext{Binomial}(n=100, p=0.6)$$, what is the standard deviation?

Accepted Answer

$$\sqrt{100 	imes 0.6 	imes 0.4}$$

Question 11

A bus arrives at a uniformly random time from 0 to 20 minutes. What is the probability that you wait between 4 and 9 minutes?

Accepted Answer

$$\frac{9-4}{20} = \frac{5}{20} = 0.25$$

Question 12

For a standard normal variable $$Z$$, what is the probability that $$Z is $$between $$-1$$ and $$1$$?

Accepted Answer

$$0.6827$$

Question 13

Exam scores are normally distributed with $$\mu = 70$$ and $$\sigma = 10. $$What is the z-score for a score of $$85$$?

Accepted Answer

$$z = \frac{85 - 70}{10} = 1.5$$

Question 14

Exam scores are normal with $$\mu = 70$$ and $$\sigma = 10. $$What score marks the 90th percentile?

Accepted Answer

$$x = 70 + 1.2816 	imes 10 \approx 82.82$$

Question 15

A population has $$\mu = 50$$ and $$\sigma = 12. $$For samples of $$n = 36$$, what is the standard error of the sample mean?

Accepted Answer

$$\frac{12}{\sqrt{36}} = 2$$

Question 16

Why is the sampling distribution of the sample mean approximately normal for $$n = 36$$ even if the population is strongly skewed?

Accepted Answer

Because the central limit theorem states that for large $$n ($$such as $$n \geq 30$$), the sampling distribution of the mean is approximately normal.

Question 17

For a normal population with $$\mu = 100$$ and $$\sigma = 15$$, compare $$P(X \u003e 110)$$ for one observation with $$P(\bar{x} \u003e 110)$$ for $$n = 25. $$Which is smaller, and why?

Accepted Answer

$$P(\bar{x} \u003e 110) is $$smaller because the standard error is less than the population standard deviation.

Question 18

A random variable $$X$$ has possible values $$0, 1, 2$$ with probabilities $$0.25, 0.35, 0.45. Is $$this a valid probability distribution?

Accepted Answer

No, because the probabilities sum to $$1.05.$$

Probability, Discrete and Binomial Distributions, and Normal Distributions: Study Guide (Chapters 4–6)

Study Guide - Practice Questions

Study Guide - Flashcards