Probability Theory for Electrical Engineers: Online Quizzes and Practice

Question 1

Let X and Y be two independent random variables with P(X = 0) = P(X = 1) = 1 / 2 and P(Y = 0) = P(Y = 1) = 1 / 2. What is P(X + Y = 1)?

Accepted Answer

1 / 4

Answer

1 / 2

Answer

3 / 4

Question 2

According to the central limit theorem, what is the distribution of the sample mean of a large number of independent, identically distributed random variables with finite mean and variance?

Accepted Answer

Normal distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Exponential distribution

Question 3

Imagine flipping a fair coin three times. What's the probability of getting at least two heads?

Accepted Answer

3/4

Answer

1/2

Answer

1/4

Answer

3/8

Question 4

Which of the following is a valid probability distribution function f(x) for a continuous random variable X?

Accepted Answer

0 ≤ f(x) ≤ 1 for all x

Answer

f(x) = 0 for all x

Answer

f(x) = 1 for all x

Answer

∫[f(x) for all x] = -1

Question 5

What is the expected value of a random variable X with probability density function f(x)?

Accepted Answer

∫[x * f(x) for all x]

Answer

∑[x * f(x) for all x]

Answer

max[x * f(x) for all x]

Answer

min[x * f(x) for all x]

Question 6

What is the probability that a randomly selected value from a uniform distribution between 0 and 1 is greater than 0.75?

Accepted Answer

0.25

Answer

0

Answer

0.5

Answer

1

Question 7

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

np^2

Question 8

What is the probability that a randomly selected student from a class of 100 students has a birthday on February 29th?

Accepted Answer

1/730

Answer

1/29

Answer

1/100

Answer

1/365

Question 9

What is the probability density function of a continuous uniform distribution over the interval [a, b]?

Accepted Answer

f(x) = 1/(b - a) for a ≤ x ≤ b

Answer

f(x) = 1/(a + b) for a ≤ x ≤ b

Answer

f(x) = (b - a)/x for a ≤ x ≤ b

Answer

f(x) = (a - b)/x for a ≤ x ≤ b

Question 10

What is the expected value of a Poisson random variable with parameter λ?

Accepted Answer

λ

Answer

1 - λ

Answer

1/λ

Answer

λ²

Question 11

In probability theory, what is a probability space?

Accepted Answer

A sample space, a sigma-algebra of events, and a probability measure on the sigma-algebra.

Answer

A set of outcomes, a set of rules, and a probability distribution.

Answer

A set of experiments, a set of results, and a probability function.

Answer

A set of events, a set of probabilities, and a sample space.

Question 12

Which of these statements is **NOT** a fundamental principle of probability theory?

Accepted Answer

The probability of an event can be negative.

Answer

The probability of an event is always between 0 and 1.

Answer

The probability of the entire sample space is 1.

Answer

The probability of the union of two events is always greater than or equal to the probability of either event.

Question 13

How does the accuracy of the Central Limit Theorem's approximation of a normal distribution improve as the sample size increases?

Accepted Answer

The accuracy increases with larger sample sizes.

Answer

The accuracy decreases with larger sample sizes.

Answer

The accuracy is unaffected by sample size.

Answer

The Central Limit Theorem is only applicable to sample sizes greater than 100.

Question 14

What is the probability of rolling a '4' on a standard six-sided die?

Accepted Answer

1/6

Answer

1/4

Answer

1/3

Answer

1/5

Question 15

An electronic component has a 5% chance of failing during its lifetime. If 200 of these components are installed in a system, what is the probability that exactly 10 will fail?

Accepted Answer

0.1238

Answer

0.2

Answer

1

Answer

0.05

Question 16

Which of the following is a property of a probability distribution function?

Accepted Answer

The sum of probabilities for all possible outcomes is 1.

Answer

The expected value of a random variable is always less than 1.

Answer

The probability of an event can be negative.

Answer

The probability of getting a '6' on a dice is always 50%.

Question 17

Which of the following statements describes the central limit theorem?

Accepted Answer

As the sample size increases, the sampling distribution of means approaches a normal distribution, regardless of the population distribution shape.

Answer

The expected value of a random variable is always equal to the variance.

Answer

The probability of getting a 'heads' on a coin toss is always 50%.

Question 18

What is the probability that at least one of the events A or B occurs?

Accepted Answer

P(A) + P(B) - P(A ∩ B)

Answer

P(A) - P(B)

Answer

P(A) * P(B)

Question 19

What is the probability of an outcome that will definitely occur?

Accepted Answer

1

Answer

0

Answer

0.5

Answer

It varies depending on the outcome.

Question 20

Which of the following is a valid probability density function (PDF) for a continuous random variable X?

Accepted Answer

f(x) = (1/sqrt(2πσ^2))e^(-(x-μ)^2/(2σ^2))

Answer

f(x) = x^2

Answer

f(x) = 1/(1+x^2)

Answer

f(x) = sin(x)

Question 21

An experiment consists of tossing a fair coin twice. What is the probability of obtaining heads on the first toss and tails on the second toss?

Accepted Answer

1/4

Answer

1/2

Answer

1/16

Answer

1/8

Question 22

The expected value of a discrete random variable X is defined as ________.

Accepted Answer

E[X] = Σ(x * P(X = x))

Answer

E[X] = P(X) * Σ(x)

Answer

E[X] = Σ(P(X = x))

Answer

E[X] = x * P(X)

Question 23

Two independent events have probabilities P(A) = 0.3 and P(B) = 0.7. What is the probability that at least one of the events occurs?

Accepted Answer

0.89

Answer

0.56

Answer

0.49

Answer

0.21

Question 24

What is the probability of drawing a red card from a standard deck of 52 cards, assuming it contains 13 red cards?

Accepted Answer

13/52

Answer

1/3

Answer

1/5

Answer

1/2

Question 25

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

Accepted Answer

7/8

Answer

3/8

Answer

5/8

Question 26

A continuous random variable X follows a uniform distribution between a and b. What is its probability density function?

Accepted Answer

f(x) = 1/(b - a) for a ≤ x ≤ b

Answer

f(x) = x for a ≤ x ≤ b

Answer

f(x) = e^(-x) for a ≤ x ≤ b

Answer

f(x) = x^2 for a ≤ x ≤ b

Question 27

Let X be a normally distributed random variable with mean μ and standard deviation σ. The probability that X lies within one standard deviation of the mean is approximately:

Accepted Answer

0.68

Answer

0.75

Answer

0.5

Question 28

Identify the incorrect property of a binomial distribution:

Accepted Answer

The sum of the probabilities of all possible values of a random variable is less than 1

Answer

The trials are independent

Answer

The probability of success is constant for each trial

Question 29

A survey of 1000 people reveals that 60% prefer coffee to tea. Calculate the standard error for the sample proportion.

Accepted Answer

0.0141

Answer

0.1

Answer

0.05

Answer

0.02

Question 30

Construct a 95% confidence interval for the population mean given that a random sample of 50 values has a mean of 10 and a standard deviation of 2.

Accepted Answer

(9.426, 10.574)

Answer

(9.300, 10.700)

Answer

(9.500, 10.500)

Question 31

Identify a limitation of probability theory:

Accepted Answer

It does not account for uncertainty or imprecision in data

Answer

It requires large sample sizes

Answer

It assumes that all events are independent

Question 32

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

Accepted Answer

1/2

Answer

1/4

Answer

1/6

Answer

1/3

Question 33

A bag contains 5 red balls and 3 blue balls. If two balls are drawn without replacement, what is the probability of drawing a red ball followed by a blue ball?

Accepted Answer

15/56

Answer

1/12

Answer

1/4

Answer

1/6

Question 34

Let X and Y be independent random variables with binomial distributions B(2, 1/4) and B(3, 1/2), respectively. What is the probability that X = 1 and Y > 1?

Accepted Answer

3/64

Answer

1/16

Answer

1/32

Answer

1/8

Question 35

A random variable X follows a normal distribution with a mean of 50 and a standard deviation of 5. What is the probability that X takes a value between 40 and 60?

Accepted Answer

0.6827

Answer

0.75

Answer

0.8

Answer

0.5

Question 36

Which of the following is NOT a property of a probability density function (PDF)?

Accepted Answer

It can have negative values.

Answer

It is a function.

Answer

It is non-negative at every point.

Answer

It integrates to 1 over the entire real line.

Question 37

A random variable X has a mean of 5 and a standard deviation of 2. Assuming X follows a normal distribution, what is the approximate probability that X is greater than 7?

Accepted Answer

0.1587

Answer

0.25

Answer

0.5

Answer

0.75

Question 38

Two events A and B are independent if:

Accepted Answer

P(A ∩ B) = P(A) · P(B).

Answer

P(B | A) = P(B).

Answer

P(A | B) = P(A).

Answer

None of the above.

Question 39

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

Accepted Answer

2/45

Answer

1/12

Answer

1/6

Answer

1/10

Question 40

Which of the following is an example of a continuous random variable that can take any value in a given interval?

Accepted Answer

Temperature in a certain region over a week.

Answer

Number of students in a class.

Answer

Outcome of a dice roll.

Answer

Number of heads obtained in 10 coin tosses.

Question 41

The joint probability distribution of two random variables X and Y is given by P(X = x, Y = y) = 1/4 for x = 1, 2 and y = 1, 2. What is the marginal probability distribution of X?

Accepted Answer

P(X = 1) = 1/2, P(X = 2) = 1/2

Answer

P(X = 1) = 1/4, P(X = 2) = 3/4

Answer

P(X = 1) = 3/4, P(X = 2) = 1/4

Answer

P(X = 1) = 1/8, P(X = 2) = 7/8

Question 42

A fair coin is flipped three times. What is the probability of getting exactly two heads?

Accepted Answer

3/8

Answer

1/8

Answer

1/2

Answer

1/4

Question 43

A random variable X follows a normal distribution with a mean of 10 and a standard deviation of 2. What is the probability that X is greater than 12?

Accepted Answer

0.1587

Answer

0.6587

Answer

0.3413

Question 44

A company produces resistors with a mean resistance of 100 ohms and a standard deviation of 5 ohms. What is the probability that a randomly selected resistor will have a resistance between 95 ohms and 105 ohms?

Accepted Answer

0.6827

Answer

0.9545

Answer

0.1587

Answer

0.3413

Question 45

Two events A and B are independent. If P(A) = 0.4 and P(B) = 0.6, what is P(A ∩ B)?

Accepted Answer

0.24

Answer

0.4

Answer

0.1

Answer

0.6

Question 46

A random variable X has a cumulative distribution function (CDF) given by F(x) = x^2 for 0 ≤ x ≤ 1. What is the probability that X is less than 0.5?

Accepted Answer

0.25

Answer

0.75

Answer

0.5

Answer

1

Question 47

A system consists of two components, each with a reliability of 0.9. The components operate independently. What is the reliability of the entire system if the components are connected in series?

Accepted Answer

0.81

Answer

1.8

Answer

0.9

Answer

0.99

Question 48

A fair dice is rolled 3 times. What is the probability of getting exactly two sixes?

Accepted Answer

9/216

Answer

1/4

Answer

1/8

Answer

1/2

Question 49

A random variable X follows a normal distribution with a mean of 50 and a standard deviation of 10.  What is the probability that X is greater than 65?

Accepted Answer

0.0668

Answer

0.8413

Answer

0.1587

Answer

0.9332

Question 50

A company manufactures resistors with resistance following a normal distribution with a mean of 100 ohms and a standard deviation of 5 ohms. What percentage of resistors will have a resistance value between 95 ohms and 105 ohms?

Accepted Answer

68.27%

Answer

34.13%

Answer

95.45%

Answer

15.87%

Question 51

A system consists of three components connected in series. The probabilities of each component failing are 0.1, 0.2, and 0.3, respectively. What is the probability that the entire system will function correctly?

Accepted Answer

0.504

Answer

0.4

Answer

0.9

Answer

0.6

Question 52

A fair six-sided die is rolled. What is the expected value of the number rolled?

Accepted Answer

3.5

Answer

3

Answer

4

Answer

6