Probability Theory for Mechanical Engineering: Interactive Learning Quizzes

Question 1

A fair coin is tossed twice. What is the probability of getting heads on the first toss and tails on the second toss?

Accepted Answer

1/4

Answer

1/2

Answer

1/3

Answer

1/6

Question 2

A bag contains 5 red balls and 3 blue balls. If 2 balls are drawn at random without replacement, what is the probability of drawing a red ball first and then a blue ball?

Accepted Answer

5/8

Answer

3/8

Answer

1/4

Answer

5/12

Question 3

A manufacturing process produces 95% acceptable products. If 5 products are randomly selected, what is the probability that at least one is defective?

Accepted Answer

0.59

Answer

0.05

Answer

0.23

Answer

0.41

Question 4

When rolling a standard six-sided die, what is the probability of obtaining a number greater than 4?

Accepted Answer

1/3

Answer

1/2

Answer

1/4

Answer

1/6

Question 5

Two fair coins are flipped. What is the probability of obtaining at least one head?

Accepted Answer

3/4

Answer

1/2

Answer

1/4

Answer

1/8

Question 6

A medical test has a 90% accuracy rate. If the probability of having the disease is 20%, what is the probability of testing positive given that the person actually has the disease?

Accepted Answer

0.82

Answer

0.18

Answer

0.90

Answer

1.00

Question 7

In a manufacturing process, the probability of producing a defective product is 0.05. If 1000 products are manufactured, what is the expected number of defective products?

Accepted Answer

5

Answer

50

Answer

500

Answer

1000

Question 8

A bag contains 3 red balls, 2 blue balls, and 1 green ball. If two balls are drawn randomly with replacement, what is the probability of drawing a red ball and then a blue ball?

Accepted Answer

9/25

Answer

1/5

Answer

1/2

Answer

2/5

Question 9

In a fair coin toss, what is the probability of getting a head?

Accepted Answer

1/2

Answer

1/4

Answer

3/4

Answer

1

Question 10

Two events, A and B, are mutually exclusive. If P(A) = 0.3 and P(B) = 0.4, what is P(A ∪ B)?

Accepted Answer

0.7

Answer

0.1

Answer

0.5

Answer

0.8

Question 11

In a standard deck of 52 playing cards, what is the probability of drawing a red card or an ace?

Accepted Answer

2/17

Answer

1/13

Answer

1/17

Answer

2/13

Question 12

Let A and B be two events with P(A) = 0.5 and P(B) = 0.7. If P(A ∩ B) = 0.3, what is P(A|B)?

Accepted Answer

2/3

Answer

1/3

Answer

1/2

Answer

5/6

Question 13

A continuous random variable X has the following probability density function:
    f(x) = 2x for 0 ≤ x ≤ 1. What is the probability that X is between 0.25 and 0.75?

Accepted Answer

0.375

Answer

0.125

Answer

0.25

Answer

0.5

Question 14

Two events, A and B, are independent if:

Accepted Answer

the occurrence of one event does not affect the probability of the other event.

Answer

the occurrence of one event affects the probability of the other event.

Answer

the probability of both events occurring simultaneously is equal to the sum of their individual probabilities.

Answer

the probability of either event occurring is less than the probability of both events occurring simultaneously.

Question 15

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If two balls are drawn at random, without replacement, what is the probability of drawing a red ball first and then a green ball?

Accepted Answer

1/15

Answer

1/10

Answer

1/6

Answer

1/5

Question 16

A machine produces widgets with a 5% probability of being defective. If a random sample of 10 widgets is inspected, what is the probability that at least 2 of them are defective?

Accepted Answer

0.309

Answer

0.107

Answer

0.206

Answer

0.412

Question 17

Which of the following probability distributions is used to model the time between arrivals in a queuing system?

Accepted Answer

Exponential distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Normal distribution

Question 18

Which formula calculates the probability of event A given that event B has already occurred?

Accepted Answer

P(A) / P(B)

Answer

P(A) * P(B)

Answer

P(B) / P(A)

Answer

P(B) * P(A)

Question 19

What is the mean of a Poisson distribution with rate parameter λ?

Accepted Answer

λ

Answer

1/λ

Answer

λ^2

Answer

1/λ^2

Question 20

What is the primary purpose of Bayes' theorem?

Accepted Answer

To calculate the probability of an event given that another event has occurred

Answer

To determine the probability of multiple independent events occurring

Answer

To predict future events based on past data

Answer

To estimate the parameters of a probability distribution

Question 21

Which assumption is key to the central limit theorem?

Accepted Answer

The sample size is large.

Answer

The population distribution is normal.

Answer

The events are independent.

Answer

The probability of success is constant.

Question 22

A fair coin is tossed three times. What is the probability of obtaining at least one head?

Accepted Answer

7 in 8

Answer

1 in 2

Answer

1 in 4

Answer

3 in 4

Question 23

A random variable X has the following probability distribution function: P(X=1) = 0.2, P(X=2) = 0.3, P(X=3) = 0.5. Calculate the expected value of X.

Accepted Answer

2.2

Answer

1.8

Answer

2.7

Answer

3.0

Question 24

Which of the following is NOT a fundamental property of probability?

Accepted Answer

Probability of an event can be negative.

Answer

Probabilities are non-negative.

Answer

The sum of probabilities of all possible outcomes is 1.

Answer

Probability of the complement of an event is 1 minus the probability of the event.

Question 25

Bayes' theorem is primarily used for calculating:

Accepted Answer

The probability of an event based on known conditions.

Answer

The expected value of a random variable.

Answer

The variance of a probability distribution.

Answer

The correlation between two random variables.

Question 26

Which of the following is a continuous probability distribution?

Accepted Answer

Exponential distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Hypergeometric distribution

Question 27

Which of the following is a valid probability mass function (PMF)?

Accepted Answer

P(x) = 0.6 for x = 1 and P(x) = 0.4 for x = 2

Answer

P(x) = 0.5 for all x

Answer

P(x) = 1/x for x = 1, 2, 3

Answer

P(x) = -0.2 for x = 1 and P(x) = 0.8 for x = 2

Question 28

A bag contains 4 red marbles and 6 blue marbles. If two marbles are drawn with replacement, what is the probability of drawing a red marble on both draws?

Accepted Answer

4/9

Answer

1/3

Answer

2/9

Answer

8/9

Question 29

Bayes' theorem is used to calculate:

Accepted Answer

The probability of a hypothesis given some evidence.

Answer

The probability of evidence given a hypothesis.

Answer

The joint probability of a hypothesis and some evidence.

Answer

The conditional probability of one hypothesis given another hypothesis.

Question 30

What is the probability of rolling a 6 on a standard six-sided die?

Accepted Answer

1/6

Answer

1/5

Answer

1/4

Answer

1/3

Question 31

What is the probability of drawing two aces in a row from a standard deck of 52 playing cards?

Accepted Answer

1/2704

Answer

1/169

Answer

1/84

Answer

1/42

Question 32

What is the likelihood ratio used in Bayes' theorem?

Accepted Answer

P(A|B) / P(B|A)

Answer

P(B|A) / P(not-B|A)

Answer

P(B|A) / P(A|B)

Answer

P(not-B|A) / P(B|A)

Question 33

Which of the following is a characteristic of a fair die?

Accepted Answer

Each side has an equal probability of landing face up when rolled.

Answer

The sum of the numbers on opposite sides is always 7.

Answer

The die is weighted so that some sides are more likely to land face up.

Answer

The die is cubic in shape.

Question 34

What is the probability of obtaining at least one head when flipping a coin three times?

Accepted Answer

3/4

Answer

1/4

Answer

1/2

Answer

1

Question 35

By the Central Limit Theorem, if X is a random sample of size n taken from a population with mean μ and standard deviation σ, what will be the distribution of the sample mean?

Accepted Answer

Normal distribution with mean μ and standard deviation σ/√n

Answer

Normal distribution with mean μ and standard deviation σ

Answer

Binomial distribution with mean μ and standard deviation σ

Answer

Binomial distribution with mean μ and standard deviation σ/√n

Question 36

What is the main difference between a prior probability and a posterior probability?

Accepted Answer

A prior probability is based on no information, while a posterior probability is based on some information.

Answer

A prior probability is based on some information, while a posterior probability is based on no information.

Answer

A prior probability is based on experimental data, while a posterior probability is based on theoretical assumptions.

Answer

A prior probability is based on theoretical assumptions, while a posterior probability is based on experimental data.

Question 37

What is the probability of rolling an even number on a fair six-sided die?

Accepted Answer

1/2

Answer

1/6

Answer

1/3

Answer

1

Question 38

What is the mean of a Poisson distribution with parameter $\lambda$?

Accepted Answer

$\lambda$

Answer

0

Answer

$\lambda^2$

Answer

1

Question 39

Which of the following axioms of probability is violated by the statement 'The probability of an event is -0.5'?

Accepted Answer

Non-negativity

Answer

Additivity

Answer

Normalization

Answer

Mutually exclusive events

Question 40

What is the formula for computing the conditional probability of event A given event B, denoted as P(A|B)?

Accepted Answer

P(A ∩ B) / P(B)

Answer

P(B) / P(A)

Answer

P(A ∪ B) / P(A)

Answer

P(A) / P(B)

Question 41

Which of the following is Bayes' theorem?

Accepted Answer

P(A|B) = P(B|A)P(A) / P(B)

Answer

P(A) = P(B|A)P(A|B) / P(B)

Answer

P(A) + P(B) = P(A ∪ B)

Answer

P(A) - P(B) = P(A ∩ B)

Question 42

Which of the following is a continuous probability distribution?

Accepted Answer

Uniform distribution

Answer

Binomial distribution

Answer

Geometric distribution

Answer

Poisson distribution

Question 43

What is the variance of a binomial distribution with parameters n and p?

Accepted Answer

np(1-p)

Answer

np

Answer

n(1-p)

Answer

p(1-p)

Question 44

Which of the following is NOT an application of probability theory in mechanical engineering?

Accepted Answer

Thermodynamics

Answer

Reliability analysis

Answer

Fatigue analysis

Answer

Vibration analysis

Question 45

What is the probability of rolling a 6 on a fair six-sided die?

Accepted Answer

1/6

Answer

1/3

Answer

1/4

Answer

1/2

Question 46

If the probability of event A is 0.4 and the probability of event B is 0.6, what is the probability that either event will occur?

Accepted Answer

0.8

Answer

0.1

Answer

0.6

Answer

1.0

Question 47

What is the mean of a normal distribution with a mean of 50 and a standard deviation of 10?

Accepted Answer

50

Answer

35

Answer

65

Answer

80

Question 48

What is the probability that a randomly selected value from a normal distribution with a mean of 100 and a standard deviation of 15 will be between 85 and 115?

Accepted Answer

0.68

Answer

0.34

Answer

0.95

Answer

1.00

Question 49

Which of the following is the formula for combinations?

Accepted Answer

nCr = n! / (r! * (n-r)!)

Answer

nPr = n! / (n-r)!

Answer

n! = n * (n-1)!

Answer

nCr = n * (n-1)

Question 50

A coin is tossed twice. What is the probability of getting a head on the first toss and a tail on the second toss?

Accepted Answer

1/4

Answer

1/3

Answer

1/2

Answer

1/8

Question 51

The probability of an event is defined as:

Accepted Answer

The ratio of favorable outcomes to total possible outcomes

Answer

The ratio of unfavorable outcomes to total possible outcomes

Answer

The difference between favorable and unfavorable outcomes

Answer

The sum of all possible outcomes

Question 52

An example of an application of Bayes' theorem is:

Accepted Answer

Predicting the probability of a disease based on symptoms

Answer

Estimating the reliability of a system

Answer

Calculating the probability of a random variable

Answer

Determining the distribution of a population

Question 53

The mean of a uniform distribution over the interval [a, b] is:

Accepted Answer

(a + b) / 2

Answer

a

Answer

b

Answer

(a - b) / 2

Question 54

A bag contains 6 red balls, 4 blue balls, and 2 green balls. If two balls are drawn randomly without replacement, what is the probability of drawing a red ball and then a blue ball?

Accepted Answer

12/35

Answer

1/5

Answer

2/15

Answer

6/35

Question 55

What is the probability density function (PDF) of a uniform distribution over the interval $$a, b$$?

Accepted Answer

f(x) = 1/(b-a) for a ≤ x ≤ b

Answer

f(x) = x/(b-a) for a ≤ x ≤ b

Answer

f(x) = 1/x

Answer

f(x) = (x-a)/(b-a) for a ≤ x ≤ b