Question 1

A random process is defined as:

Accepted Answer

A process that generates results that are determined by chance, and the outcome cannot be predicted in advance.

Question 2

Suppose you flip a fair coin 100 times. Which of the following best describes the Law of Large Numbers in this context?

Accepted Answer

As the number of flips increases, the proportion of heads will get closer to the true probability of 0.5.

Question 3

If $$P(A) = 0.4$$, $$P(B) = 0.6$$, and $$P(A \cap B) = 0.24$$, are events A and B independent?

Accepted Answer

Yes, because $$P(A \cap B) = P(A) \cdot P(B).$$

Question 4

A probability distribution for a discrete random variable X must satisfy which two properties?

Accepted Answer

Each probability is between 0 and 1, and the sum of all probabilities is 1.

Question 5

If events A and B are mutually exclusive, what is $$P(A \cap B)$$?

Accepted Answer

0

Question 6

A bag contains 5 red and 5 blue marbles. Two marbles are drawn at random without replacement. What is the probability that both marbles are red?

Accepted Answer

$$\dfrac{5}{10} \cdot \dfrac{4}{9}$$

Question 7

Given a probability distribution for a random variable X, the mean (expected value) is calculated as:

Accepted Answer

$$\mu_X = \sum x \cdot P(x)$$

Question 8

Suppose you roll a fair six-sided die. What is the probability of rolling an odd number or a prime number?

Accepted Answer

$$\dfrac{5}{6}$$

Question 9

If $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.1$$, what is $$P(A \cup B)$$?

Accepted Answer

$$0.3 + 0.5 - 0.1 = 0.7$$

Question 10

A random variable X has the following probability distribution: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$What is $$P(X \leq 2)$$?

Accepted Answer

$$0.7$$

Question 11

A binomial random variable is defined by which of the following characteristics?

Accepted Answer

A fixed number of trials, each with two possible outcomes, and the probability of success is the same for each trial.

Question 12

If $$X is a $$binomial random variable with $$n=10$$ and $$p=0.3$$, what is the expected value $$E[X]$$?

Accepted Answer

$$3$$

Question 13

Suppose the probability that a randomly selected student is happy is $$0.612$$ and the probability that a student is happy and free is $$0.145. $$What is the conditional probability that a student is free given they are happy?

Accepted Answer

$$\dfrac{0.145}{0.612}$$

Question 14

If $$X$$ and $$Y$$ are independent random variables with $$\sigma_X = 2$$ and $$\sigma_Y = 3$$, what is the standard deviation of $$X + Y$$?

Accepted Answer

$$\sqrt{2^2$ + 3^2$} = \sqrt{13}$$

Question 15

A linear transformation is applied to a random variable $$X$$ such that $$Y = 3X + 2. If$$ $$\mu_X = 4$$ and $$\sigma_X = 1.5$$, what are $$\mu_Y$$ and $$\sigma_Y$$?

Accepted Answer

$$\mu_Y = 14$$, $$\sigma_Y = 4.5$$

Question 16

Which of the following is NOT a property of a probability distribution for a discrete random variable?

Accepted Answer

Probabilities can be negative as long as they sum to 1.

Question 17

If $$P(A) = 0.7$$, what is the probability of the complement of event A?

Accepted Answer

$$0.3$$

Question 18

A simulation is used to estimate the probability of a rare event. Which of the following best describes the advantage of using simulation?

Accepted Answer

It allows us to model complex random processes that are difficult to analyze mathematically.

Question 19

Suppose you want to find the probability that a randomly selected insurance policy results in a profit of $$100 for the company. The probability distribution is: P(100) = 0.8$, P(-150) = 0.15$, P(-3500) = 0.05$. What is the expected profit?

Accepted Answer

$$0.8 	imes 100 + 0.15 	imes (-150) + 0.05 	imes (-3500) = -115$$

AP Statistics: Probability, Random Variables, and Binomial Distributions Study Guide

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