Probability in Calculus I: Interactive Learning Quizzes

Question 1

Which integral represents the probability of selecting a random number from a continuous distribution with pdf f(x) over the interval [a, b]?

Accepted Answer

∫[a,b] f(x) dx

Answer

∫[0,1] f(x) dx

Answer

∫[-∞,∞] f(x) dx

Answer

∫[a,∞] f(x) dx

Question 2

What is the probability of choosing a random point inside a circle with radius r centered at the origin?

Accepted Answer

πr²/4

Answer

r²/4

Answer

r²/2π

Answer

2πr²

Question 3

Which of the following is a crucial assumption in using integration to calculate probabilities?

Accepted Answer

The sample space is continuous.

Answer

The event is independent of other events.

Answer

The probability is uniformly distributed.

Answer

The event is discrete.

Question 4

Which expression represents the probability of a random variable X taking on a value between a and b, considering its pdf f(x)?

Accepted Answer

∫[a,b] f(x) dx

Answer

f(x) dx

Answer

∫[0,a] f(x) dx

Answer

∫[b,∞] f(x) dx

Question 5

Given a random variable X with a variance of 16 and a mean of 5, what is the standard deviation of X?

Accepted Answer

4

Answer

2

Answer

8

Answer

16

Question 6

Which of the following is the probability mass function of a binomial distribution with n trials and success probability p?

Accepted Answer

P(X = x) = (n! / (x! * (n - x)!)) * p^x * (1 - p)^(n - x)

Answer

P(X ≤ x) = (n! / (x! * (n - x)!)) * p^x * (1 - p)^(n - x)

Answer

E(X) = n * p

Answer

Var(X) = n * p * (1 - p)

Question 7

In medical diagnostics, how is probability used to determine the likelihood of a patient having a particular condition given a positive test result?

Accepted Answer

Using Bayes' Theorem

Answer

Using the binomial distribution

Answer

Using the Poisson distribution

Answer

Using the normal distribution

Question 8

Given a random variable X with a probability density function f(x) = 1/2 for 0 ≤ x ≤ 2, determine the probability of X being less than 1.

Accepted Answer

1/4

Answer

1/2

Answer

3/4

Answer

1

Question 9

Consider two independent events A and B with probabilities P(A) = 0.4 and P(B) = 0.6. What is the probability of either event occurring?

Accepted Answer

0.8

Answer

0.2

Answer

0.64

Answer

0.36

Question 10

Which of the following is the probability density function of the standard normal distribution?

Accepted Answer

f(x) = (1 / √(2π)) * e^(-x^2 / 2)

Answer

f(x) = (1 / √(2π)) * e^(-x / 2)

Answer

f(x) = (2 / √(π)) * e^(-x^2 / 2)

Question 11

Which of the following is NOT an application of probability theory?

Accepted Answer

Predicting the weather with certainty

Answer

Evaluating medical test results

Answer

Assessing reliability of systems

Answer

Modeling financial markets

Question 12

Which of the following is a key concept in Bayesian probability?

Accepted Answer

Updating probabilities based on new evidence

Answer

Assuming all events are equally likely

Answer

Using only objective data

Answer

Ignoring prior information

Question 13

When rolling a six-sided die, what is the probability of rolling an even number?

Accepted Answer

1/2

Answer

1/3

Answer

2/3

Answer

1

Question 14

A biased coin has a probability of landing on heads of 0.7. If the coin is flipped 10 times, what is the probability of getting exactly 6 heads?

Accepted Answer

0.168

Answer

0.056

Answer

0.7

Answer

0.224

Question 15

Let X be a random variable with a mean of 5 and a standard deviation of 2. What is the probability that X is less than 2?

Accepted Answer

0.0228

Answer

0.9772

Answer

0.25

Answer

0.5

Question 16

Which of the following is a postulate of probability theory?

Accepted Answer

The probability of the union of any two events is less than or equal to the sum of their individual probabilities

Answer

The sum of probabilities of all possible outcomes in a sample space is equal to 0

Answer

The probability of an impossible event is greater than 0

Question 17

Which of the following is an example of a discrete random variable?

Accepted Answer

Number of heads when flipping a coin 5 times.

Answer

Height of a randomly selected person.

Answer

Temperature outside.

Answer

Speed of a moving car.

Question 18

Which of the following is an application of probability in real life?

Accepted Answer

Predicting weather patterns.

Answer

Finding the area of a circle.

Answer

Solving algebraic equations.

Answer

Calculating the volume of a sphere.

Question 19

A fair 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

Answer

3/4

Answer

1/2

Question 20

A quality control manager randomly selects 10 items from a production line. If the probability of a defective item is 0.1, what is the probability that there are exactly 2 defective items?

Accepted Answer

0.2436

Answer

0.3536

Answer

0.4961

Answer

0.1353

Question 21

What is the formula for calculating the expected value (mean) of a continuous random variable X with probability density function f(x)?

Accepted Answer

E(X) = ∫xf(x) dx

Answer

E(X) = ∫f(x)² dx

Answer

E(X) = ∫x²f(x) dx

Answer

E(X) = ∫f(x) dx

Question 22

A random variable X is uniformly distributed over the interval [0, 2]. What is its probability density function, f(x)?

Accepted Answer

f(x) = 1/2 for 0 ≤ x ≤ 2

Answer

f(x) = 1/x for 0 ≤ x ≤ 2

Answer

f(x) = x for 0 ≤ x ≤ 2

Answer

f(x) = 2 for 0 ≤ x ≤ 2

Question 23

How is the standard deviation of a continuous random variable X related to its variance?

Accepted Answer

The standard deviation is the square root of the variance.

Answer

The standard deviation is the square of the variance.

Answer

The standard deviation is the variance minus the expected value.

Answer

The standard deviation is the expected value minus the variance.

Question 24

What is the probability density function, f(x), of an exponential distribution with parameter λ?

Accepted Answer

f(x) = λe^(-λx) for x ≥ 0

Answer

f(x) = λe^(λx) for x ≥ 0

Answer

f(x) = e^(-λx) for x ≥ 0

Answer

f(x) = e^(λx) for x ≥ 0

Question 25

What does the area under the curve of a probability density function represent?

Accepted Answer

The total probability of the random variable (which is always 1)

Answer

The standard deviation of the random variable

Answer

The variance of the random variable

Answer

The expected value of the random variable

Question 26

A continuous random variable X has a probability density function f(x) = 1/x for 1 ≤ x ≤ e.  What is the approximate variance of X?

Accepted Answer

Approximately 0.231

Answer

Approximately 0.75

Answer

Approximately 1

Answer

Approximately 0.5

Question 27

Let X be a continuous random variable with probability density function f(x) = 2x for 0 ≤ x ≤ 1. Find the probability that X is less than or equal to 0.5.

Accepted Answer

0.25

Answer

0.5

Answer

0.75

Answer

1

Question 28

Let X be a random variable with a uniform distribution on the interval [0, 2]. Calculate the expected value of X.

Accepted Answer

1

Answer

0.5

Answer

0

Answer

2

Question 29

Given the probability density function of a random variable X as f(x) = (1/2)e^(-x/2) for x ≥ 0, determine the cumulative distribution function (CDF) of X.

Accepted Answer

F(x) = 1-e^(-x/2)

Answer

F(x) = (1/2)e^(-x/2)

Answer

F(x) = e^(-x/2)

Answer

F(x) = 1+e^(-x/2)

Question 30

A fair coin is tossed 4 times. Find the probability of obtaining at least 3 heads.

Accepted Answer

5/16

Answer

1/16

Answer

1/2

Question 31

A continuous random variable X has a probability density function f(x) = kx for 0 ≤ x ≤ 2, where k is a constant. Determine the value of k.

Accepted Answer

1/2

Answer

2

Answer

1

Answer

4

Question 32

The lifetime of a certain type of light bulb follows an exponential distribution with a mean of 1000 hours. Calculate the probability that a randomly selected bulb lasts more than 1500 hours.

Accepted Answer

e^(-1.5)

Answer

e^(-0.5)

Answer

e^(-2)

Answer

e^(-1)

Question 33

A normally distributed random variable X has a standard deviation of 5. Find the probability that X is within one standard deviation of its mean.

Accepted Answer

Approximately 0.68

Answer

Approximately 0.95

Answer

Approximately 0.5

Answer

Approximately 0.997

Question 34

A company manufactures light bulbs with a mean lifespan of 1000 hours and a standard deviation of 100 hours, assuming they are normally distributed. Determine the probability that a randomly selected bulb will last between 900 and 1100 hours.

Accepted Answer

Approximately 0.68

Answer

Approximately 0.5

Answer

Approximately 0.95

Answer

Approximately 0.997

Question 35

Given the probability density function of a random variable X as f(x) = 1/4 for 0 ≤ x ≤ 4, find the variance of X.

Accepted Answer

4/3

Answer

1/3

Answer

1

Answer

2

Question 36

Let X be a random variable with probability density function f(x) = (1/10)x for 0 ≤ x ≤ 10. Determine the median of X.

Accepted Answer

√10

Answer

5

Answer

2√10

Answer

10

Question 37

A continuous random variable X has a probability density function f(x) = 2x for 0 ≤ x ≤ 1 and 0 otherwise. What is the probability that X is greater than 0.5?

Accepted Answer

0.75

Answer

1

Answer

0.25

Answer

0.5

Question 38

The lifetime of a certain type of light bulb is exponentially distributed with a mean of 1000 hours. What is the probability that a randomly selected light bulb will last more than 1500 hours?

Accepted Answer

e^(-1.5)

Answer

1 - e^(-1.5)

Answer

e^(-0.5)

Answer

1 - e^(-0.5)

Question 39

A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?

Accepted Answer

10/32

Answer

5/32

Answer

1/32

Answer

1/2

Question 40

A random variable X has a uniform distribution on the interval [0, 2]. What is the expected value of X?

Accepted Answer

1

Answer

0.5

Answer

0

Answer

2

Question 41

The probability density function of a continuous random variable X is given by f(x) = kx^2 for 0 ≤ x ≤ 1 and 0 otherwise. What is the value of k?

Accepted Answer

3

Answer

1

Answer

1/3

Answer

1/2

Question 42

A company produces light bulbs with a mean lifetime of 1000 hours and a standard deviation of 100 hours. Assuming the lifetime is normally distributed, what is the probability that a randomly selected light bulb will last between 900 and 1100 hours?

Accepted Answer

Approximately 0.6827

Answer

Approximately 0.9545

Answer

Approximately 0.1587

Answer

Approximately 0.3413

Question 43

A random variable X has a probability density function f(x) = 1/2 for 0 ≤ x ≤ 2 and 0 otherwise. What is the variance of X?

Accepted Answer

1/3

Answer

1/2

Answer

2/3

Answer

1

Question 44

A box contains 5 red balls and 3 blue balls. Two balls are drawn at random without replacement. What is the probability that both balls are red?

Accepted Answer

5/14

Answer

3/8

Answer

5/8

Answer

1/2

Question 45

A continuous random variable X has a probability density function f(x) = (1/2)e^(-x/2) for x ≥ 0 and 0 otherwise. What is the cumulative distribution function F(x) for X?

Accepted Answer

1 - e^(-x/2)

Answer

x/2 - e^(-x/2)

Answer

e^(-x/2)

Question 46

A random variable X has a mean of 5 and a standard deviation of 2. What is the probability that X is within one standard deviation of its mean?

Accepted Answer

Cannot be determined with the given information.

Answer

Approximately 0.9545

Answer

Approximately 0.6827

Answer

Approximately 0.3413

Question 47

The probability of drawing a king or a queen from a standard deck of 52 playing cards is:

Accepted Answer

1/2

Answer

1/8

Answer

1/4

Answer

3/8

Question 48

The probability that a random variable X takes on a value between 0 and 1 is:

Accepted Answer

1

Answer

0

Answer

Cannot be determined

Answer

0.5

Question 49

Which of the following is a continuous probability distribution?

Accepted Answer

Normal distribution

Answer

Hypergeometric distribution

Answer

Binomial distribution

Answer

Poisson distribution

Question 50

The expected value of a random variable X with probability mass function P(X = x) = 0.2 for x = 1, 2, 3, and 4 is:

Accepted Answer

2.6

Answer

1.5

Answer

3.2

Answer

2.1

Question 51

If the variance of a random variable X is 0, then X is a:

Accepted Answer

Constant random variable

Answer

Discrete random variable

Answer

Continuous random variable

Question 52

The probability density function of a continuous random variable X is f(x) = 2x for 0 ≤ x ≤ 1. The cumulative distribution function of X is:

Accepted Answer

F(x) = x² for 0 ≤ x ≤ 1

Answer

F(x) = 2 - x² for 0 ≤ x ≤ 1

Answer

F(x) = 1 - x² for 0 ≤ x ≤ 1

Answer

F(x) = 2x for 0 ≤ x ≤ 1

Question 53

If X and Y are independent random variables, then the covariance between X and Y is:

Accepted Answer

0

Answer

-1

Answer

1

Answer

Cannot be determined

Question 54

The correlation coefficient between two random variables X and Y can take values between:

Accepted Answer

-1 and 1

Answer

-1 and 0

Answer

0 and +∞

Answer

0 and 1

Question 55

Which of the following is a measure of the strength of the linear relationship between two random variables?

Accepted Answer

Correlation coefficient

Answer

Covariance

Answer

Mean

Answer

Standard deviation

Question 56

If the joint probability distribution of two random variables X and Y is given by P(X = x, Y = y) = xy for x = 1, 2 and y = 1, 2, then the covariance between X and Y is:

Accepted Answer

1/2

Answer

-1/2

Answer

0

Answer

1