Probability Theory for Engineering: Interactive Quizzes and Practice

Question 1

Which property does NOT apply to a probability space?

Accepted Answer

Probability is a subjective assessment.

Answer

Event probabilities are between 0 and 1.

Answer

The sum of all event probabilities is 1.

Answer

Event probabilities are proportional to areas in the sample space.

Question 2

The mean of a random variable X with probability mass function P(X=x) is defined as:

Accepted Answer

E(X) = Σx * P(X=x)

Answer

E(X) = ΣP(X=x)

Answer

E(X) = Σx^2 * P(X=x)

Answer

E(X) = Σ(x-μ)^2 * P(X=x)

Question 3

Which is NOT a characteristic of a Poisson distribution?

Accepted Answer

Has a skewed distribution.

Answer

Models rare events in a fixed interval.

Answer

Mean and variance are equal.

Answer

Probability of k occurrences is e^-λ * λ^k / k!

Question 4

Which of the following is NOT a type of joint probability distribution?

Accepted Answer

Conditional joint distribution

Answer

Discrete joint distribution

Answer

Continuous joint distribution

Answer

Mixed joint distribution

Question 5

Which of the following statements is NOT an axiom of probability theory?

Accepted Answer

The probability of an event is greater than or equal to -1.

Answer

The probability of an event cannot be negative.

Answer

The probability of the sample space is 1.

Answer

The probability of the union of two disjoint events is the sum of their probabilities.

Question 6

Which of the following is a discrete probability distribution?

Accepted Answer

Binomial distribution

Answer

Normal distribution

Answer

Exponential distribution

Answer

Uniform distribution

Question 7

Which of the following is a measure of the spread of a discrete probability distribution?

Accepted Answer

Variance

Answer

Mean

Answer

Standard deviation

Answer

Median

Question 8

Which of the following is NOT a fundamental component of a probability space?

Accepted Answer

Outcome

Answer

Sample space

Answer

Event

Answer

Probability measure

Question 9

In a probability experiment, the complement of an event A is:

Accepted Answer

The event that A does not occur

Answer

The event that A occurs

Answer

The union of A and its complement

Answer

The intersection of A and its complement

Question 10

Which of the following is NOT a sample space for rolling a fair six-sided die?

Accepted Answer

The set {1, 3, 5, 7}

Answer

The set {1, 2, 3, 4, 5, 6}

Answer

The set {even, odd}

Answer

The set {less than 4, greater than or equal to 4}

Question 11

Let X be a random variable that follows a uniform distribution over the interval [0, 1]. What is the probability density function of X?

Accepted Answer

f(x) = 1 for x in [0, 1]

Answer

f(x) = x for x in [0, 1]

Answer

f(x) = 2x for x in [0, 1]

Answer

f(x) = 1/x for x in [0, 1]

Question 12

Which statement about the Central Limit Theorem is correct?

Accepted Answer

The sample mean of a large number of independent random variables is approximately normally distributed.

Answer

The sum of a large number of independent random variables is always normally distributed.

Answer

The sample variance of a large number of independent random variables is approximately normally distributed.

Answer

The median of a large number of independent random variables is approximately normally distributed.

Question 13

A fair coin is tossed twice. What is the probability that both tosses result in heads?

Accepted Answer

0.25

Answer

0.75 (Incorrect)

Answer

0.5 (Incorrect)

Answer

1 (Incorrect)

Question 14

A bag contains 5 red balls and 3 blue balls. If two balls are drawn randomly without replacement, what is the probability that one ball is red and the other is blue?

Accepted Answer

0.3

Answer

0.6 (Incorrect)

Answer

0.5 (Incorrect)

Answer

0.4 (Incorrect)

Question 15

A random variable X has the following probability distribution:
X: 1 2 3
P(X): 0.2 0.3 0.5
What is the expected value of X?

Accepted Answer

2.3

Answer

1.8 (Incorrect)

Answer

2.5 (Incorrect)

Answer

3 (Incorrect)

Question 16

A continuous random variable has a probability density function of f(x) = 2x for 0 ≤ x ≤ 1. What is the probability that the random variable takes a value between 0.2 and 0.5?

Accepted Answer

0.3

Answer

0.2 (Incorrect)

Answer

0.5 (Incorrect)

Answer

0.4 (Incorrect)

Question 17

Two independent random variables X and Y have the following probability distributions:
X: 0 1
P(X): 0.6 0.4
Y: 2 3
P(Y): 0.7 0.3
What is the probability that X is less than 1 and Y is greater than 2?

Accepted Answer

0.24

Answer

0.32 (Incorrect)

Answer

0.28 (Incorrect)

Question 18

A random sample of 50 observations is taken from a population with a mean of 100 and a standard deviation of 15. What is the probability that the sample mean is between 95 and 105?

Accepted Answer

0.9544

Answer

0.7000 (Incorrect)

Answer

0.8000 (Incorrect)

Answer

0.9000 (Incorrect)

Question 19

A manufacturing process produces items with a defect rate of 5%. If a random sample of 100 items is inspected, what is the probability that at least one item is defective?

Accepted Answer

0.3935

Answer

0.5400 (Incorrect)

Answer

0.4600 (Incorrect)

Question 20

In a hypothesis test, the null hypothesis is rejected at a significance level of 0.05. What is the probability of committing a Type I error?

Accepted Answer

0.05

Answer

0.01 (Incorrect)

Answer

0.1 (Incorrect)

Answer

0.2 (Incorrect)

Question 21

A random variable X has a Poisson distribution with a mean of 3. What is the probability that X takes a value of exactly 4?

Accepted Answer

0.2240

Answer

0.3053 (Incorrect)

Answer

0.1833 (Incorrect)

Answer

0.2646 (Incorrect)

Question 22

In a regression model used to predict the value of a dependent variable based on one or more independent variables, what is the purpose of calculating the coefficient of determination (R-squared)?

Accepted Answer

To measure the proportion of variation in the dependent variable explained by the independent variables

Answer

To determine the significance of the independent variables (Incorrect)