Q: Suppose events $$A$$ and $$B$$ are independent with $$P(A) = 0.3$$ and $$P(B) = 0.5. $$What is $$P(A \text{ and } B)$$?

$$P(A \text{ and } B) = P(A)P(B) = 0.15$$

Q: In a large city, 40% of teens vape. If 3 teens are selected randomly, what is the probability that at least one of them does NOT vape?

$$P(\text{at least one non-vaper}) = 1 - (0.4)^3$ = 0.936$$

Q: Suppose $$P(L) = 0.25$$, $$P(M) = 0.35$$, and $$P(M|L) = 0.20. $$What is $$P(L \text{ and } M)$$?

$$P(L \text{ and } M) = P(L)P(M|L) = 0.25 \times 0.20 = 0.05$$

Q: An urn contains 5 red and 10 blue marbles. If 2 marbles are drawn without replacement, what is the probability that both are red?

$$\frac{5}{15} \times \frac{4}{14} = \frac{20}{210} = \frac{2}{21}$$

Q: Suppose $$z = 2.25$$, $$\mu = 12$$, $$\sigma = 5. $$What is the corresponding data value $$x$$?

$$x = \mu + z\sigma = 12 + 2.25 \times 5 = 23.25$$

Question 1

Suppose events $$A$$ and $$B$$ are independent with $$P(A) = 0.3$$ and $$P(B) = 0.5. $$What is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A)P(B) = 0.15$$

Question 2

In a large city, 40% of teens vape. If 3 teens are selected randomly, what is the probability that at least one of them does NOT vape?

Accepted Answer

$$P(	ext{at least one non-vaper}) = 1 - (0.4)^3$ = 0.936$$

Question 3

A fraternity with 15 brothers wants to select 5 brothers for clean-up duty. How many ways can this be done?

Accepted Answer

$$C_{15}^5 = \frac{15!}{5!10!} = 3003$$

Question 4

Suppose $$P(L) = 0.25$$, $$P(M) = 0.35$$, and $$P(M|L) = 0.20. $$What is $$P(L 	ext{ and } M)$$?

Accepted Answer

$$P(L 	ext{ and } M) = P(L)P(M|L) = 0.25 	imes 0.20 = 0.05$$

Question 5

An urn contains 5 red and 10 blue marbles. If 2 marbles are drawn without replacement, what is the probability that both are red?

Accepted Answer

$$\frac{5}{15} 	imes \frac{4}{14} = \frac{20}{210} = \frac{2}{21}$$

Question 6

Suppose $$X is a $$binomial random variable with $$n=12$$ and $$p=0.4. $$What is $$P(X=5)$$?

Accepted Answer

$$P(X=5) = C_{12}^5 (0.4)^5$ (0.6)^7$

Question 7

Given the probability distribution: $$x = 1,2,3,4,5$$ with $$P(X=x) = 0.1, 0.2, 0.2, 0.2, 0.3$$, what is the expected value $$\mu$$?

Accepted Answer

$$\mu = 1(0.1) + 2(0.2) + 3(0.2) + 4(0.2) + 5(0.3) = 3.5$$

Question 8

Suppose $$x=35$$, $$\mu=10$$, $$\sigma=5. $$What is the $$z-$$score for $$x$$?

Accepted Answer

$$z = \frac{x-\mu}{\sigma} = \frac{35-10}{5} = 5$$

Question 9

Katherine flips a biased coin 4 times. If $$P(H) = 0.6$$, what is the probability of getting all heads?

Accepted Answer

$$(0.6)^4 = 0.1296$

Question 10

An urn contains 5 red and 10 blue marbles. If 3 marbles are drawn with replacement, what is the probability that all are red?

Accepted Answer

$$(\frac{5}{15})^3$ = (\frac{1}{3})^3$ = \frac{1}{27}$$

Question 11

Suppose 40% of men have false teeth and 30% of men with false teeth also wear a toupee. What is the probability that a randomly selected man has false teeth and wears a toupee?

Accepted Answer

$$0.4 	imes 0.3 = 0.12$$

Question 12

Suppose $$z = 2.25$$, $$\mu = 12$$, $$\sigma = 5. $$What is the corresponding data value $$x$$?

Accepted Answer

$$x = \mu + z\sigma = 12 + 2.25 	imes 5 = 23.25$$

Question 13

Which of the following best describes the difference between permutations and combinations?

Accepted Answer

Permutations count arrangements where order matters; combinations count groupings where order does not matter.

Question 14

If $$P(A) = 0.2$$, $$P(B) = 0.3$$, and $$A$$ and $$B$$ are independent, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$P(A 	ext{ or } B) = P(A) + P(B) - P(A)P(B) = 0.2 + 0.3 - 0.06 = 0.44$$

Question 15

A deck has 52 cards. Ethan randomly selects 2 cards without replacement. What is the probability both are Aces?

Accepted Answer

$$\frac{4}{52} 	imes \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$$

Question 16

Suppose $$P(T 	ext{ and } C) = 0.20$$ and $$P(T) = 0.30$$ for nursing students. If a student is known to be tired, what is the probability they are also cranky?

Accepted Answer

$$P(C|T) = \frac{P(T 	ext{ and } C)}{P(T)} = \frac{0.20}{0.30} = 0.6667$$

Question 17

Given a standard normal distribution, what is $$P(Z \u003c -1.14)$$?

Accepted Answer

Approximately $$0.1271$$

Question 18

How many 5-letter words can be formed from the symbols R, S, T, U, V, W, X, Y, Z if no letter can be reused?

Accepted Answer

$$P_9^5$$ = \frac{9!}{4!} = 15120$$

Question 19

Suppose $$X is a $$binomial random variable with $$n=12$$ and $$p=0.4. $$What is $$P(X \geq 5)$$?

Accepted Answer

$$P(X \geq 5) = 1 - P(X \leq 4)$$

Probability, Counting, and Discrete Distributions: Study Notes

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