Probability Theory: Master the Fundamentals of Probability for Statistics MSc

Question 1

Which of the following is a stochastic process, as opposed to a probability distribution?

Accepted Answer

Poisson process

Answer

Uniform distribution

Answer

Normal distribution

Answer

Binomial distribution

Question 2

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

Accepted Answer

Binomial distribution

Answer

Continuous probability distribution

Answer

Discrete probability distribution

Answer

Uniform probability distribution

Question 3

In a class of 30 students, each with a different birthday, what is the probability that a randomly selected student has a birthday in January?

Accepted Answer

1/12

Answer

1/30

Answer

1/4

Answer

1/6

Question 4

Which statement does NOT define a fundamental characteristic of probability?

Accepted Answer

Probability is influenced by personal opinions and beliefs.

Answer

Probability is expressed numerically between 0 and 1.

Answer

Probability measures the likelihood of an event occurring.

Answer

Probabilities of mutually exclusive events can be added together.

Question 5

Given that events A and B are disjoint, and the probability of event A is 0.4 and the probability of event B is 0.6, what is the probability of the union of events A and B?

Accepted Answer

1.0

Answer

0.2

Answer

0.4

Answer

0.6

Question 6

Given two discrete random variables X and Y with joint probability mass function p(x, y), what is the probability of event X occurring?

Accepted Answer

p(X)

Answer

p(Y)

Answer

p(X, Y)

Answer

p(X) + p(Y)

Question 7

What is the definition of probability?

Accepted Answer

The likelihood of an event happening.

Answer

The number of possible outcomes for an event.

Answer

The average number of times an event will happen.

Answer

The probability of an event is always 0.5.

Question 8

When a fair six-sided die is rolled twice, what's the probability that the sum of the numbers rolled will be 7?

Accepted Answer

1/12

Answer

1/6

Answer

1/18

Answer

1/36

Question 9

We have a bag with 10 balls, 4 red and 6 blue. If we draw two balls without replacing them, what's the probability that both are red?

Accepted Answer

4/45

Answer

1/5

Answer

1/2

Answer

1/10

Question 10

In a coin toss experiment, where the probability of getting heads is 1/2, you toss the coin twice. What is the probability of getting tails both times?

Accepted Answer

1/4

Answer

1/2

Answer

1

Answer

0

Question 11

When flipping a fair coin twice, what's the likelihood of getting heads at least once?

Accepted Answer

75%

Answer

50%

Answer

25%

Answer

100%

Question 12

What is the definition of conditional probability?

Accepted Answer

The probability of an event occurring given that another event has already occurred.

Answer

The probability of two or more events occurring simultaneously.

Answer

The probability of an event occurring regardless of whether another event has occurred.

Answer

The probability of an event occurring without being influenced by any other events.

Question 13

In a parking lot, 60% of the cars are blue. What is the probability that a randomly selected car is not blue?

Accepted Answer

40%

Answer

30%

Answer

10%

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 at least one of the balls drawn is blue?

Accepted Answer

13/15

Answer

7/15

Answer

1/2

Question 15

Define independence in probability theory. Provide examples of independent events and explain how independence affects probability calculations.

Accepted Answer

Independence occurs when the outcome of one event does not influence the probability of another event.

Answer

Independence is only possible when the two events are not mutually exclusive.

Answer

Independent events can have any probability value between 0 and 1.

Answer

Independence is a property of events, not probability distributions.

Question 16

Which of the following statements about a probability distribution is **not** true?

Accepted Answer

The probability of an impossible event is 0.

Answer

The probability of an event is a non-negative number.

Answer

The probability of the sample space is 1.

Answer

The probability of the complement of an event is equal to 1 minus the probability of the event.

Question 17

A fair six-sided die is rolled twice. What is the probability that the sum of the two numbers rolled is greater than 7?

Accepted Answer

1/2

Answer

1/6

Answer

1/3

Answer

5/6

Question 18

Which of the following is the formula for Bayes' theorem?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 19

The expected value of a discrete random variable with probability mass function P(X = x) is:

Accepted Answer

E(X) = ∑xP(X = x)

Answer

E(X) = ∫xP(X = x)dx

Answer

E(X) = P(X > 0)

Answer

E(X) = P(X < 1)

Question 20

Bayes' theorem is used to calculate:

Accepted Answer

The probability of an event given that another event has occurred

Answer

The probability of an event given that another event has not occurred

Answer

The probability of two events occurring simultaneously

Answer

The probability of an event not occurring

Question 21

What is the probability of obtaining at least one six when rolling two fair six-sided dice?

Accepted Answer

25/36

Answer

1/6

Answer

1/4

Answer

11/36

Question 22

The variance of a random variable with mean μ and standard deviation σ is:

Accepted Answer

σ^2

Answer

μ

Answer

√σ

Answer

√μ

Question 23

The moment generating function of a random variable X is defined as:

Accepted Answer

M(t) = E(e^(tX))

Answer

M(t) = E(X^t)

Answer

M(t) = P(X = t)

Answer

M(t) = P(X > t)

Question 24

For discrete random variables X and Y with a joint probability distribution, how do you determine the marginal probability distribution of X? Describe its connection to the joint distribution.

Accepted Answer

Sum the joint probabilities of X and Y for all possible values of Y.

Answer

Subtract the joint probabilities of X and Y from the probability of the sample space.

Answer

Divide the joint probabilities of X and Y by the total number of outcomes.

Question 25

Let X and Y be two independent random variables with respective probability density functions f(x) and g(y). What is the probability density function of Z = X + Y?

Accepted Answer

f(x) * g(y)

Answer

f(x + y)

Answer

g(x + y)

Answer

f(x) + g(y)

Question 26

Which of the following is a commonly used method for estimating the probability of a rare event?

Accepted Answer

Monte Carlo simulation

Answer

Bayesian inference

Answer

Law of large numbers

Answer

Central limit theorem

Question 27

What is the expected value of a uniform distribution over the interval [a, b]?

Accepted Answer

(a + b) / 2

Answer

(a - b) / 2

Answer

(a * b) / 2

Answer

(a / b) / 2

Question 28

A random variable is a variable whose value is determined by the outcome of a random experiment. Which of the following types of random variables is discrete?

Accepted Answer

The number of defective items in a batch of products

Answer

The time until the next customer arrives at a store

Answer

The weight of a randomly selected apple

Answer

The temperature at a given time

Question 29

The expected value of a random variable is a measure of its central tendency. Which of the following is the formula for the expected value of a discrete random variable?

Accepted Answer

E(X) = Σ[x * P(X = x)]

Answer

E(X) = ∫[x * f(x)] dx

Answer

E(X) = Σ[P(X = x)]

Answer

E(X) = ∫[f(x)] dx

Question 30

The variance of a random variable measures how spread out its values are. Which of the following is the formula for the variance of a discrete random variable?

Accepted Answer

Var(X) = Σ[(X - E(X))^2 * P(X = x)]

Answer

Var(X) = E[(X - E(X))^2]

Answer

Var(X) = ∫[(X - E(X))^2 * f(x)] dx

Answer

Var(X) = Σ[P(X = x)^2]

Question 31

For a continuous random variable with probability density function $$f(x) = xe^{-x^2/2}$$, determine its cumulative distribution function and explain its significance.

Accepted Answer

$$F(x) = 1 - e^{-x^2/2}$$

Answer

$$F(x) = x^2e^{-x^2/2}$$

Answer

$$F(x) = -xe^{-x^2/2}$$

Answer

$$F(x) = e^{-x^2/2}$$

Question 32

Probability models are often used to aid decision-making. Discuss the challenges and ethical considerations associated with using probability models in decision-making. Provide real-world examples to illustrate where these issues arise.

Accepted Answer

Probability models can be a valuable tool for decision making, but it is important to be aware of their limitations and ethical implications.

Answer

Probability models are always accurate and should be used without question.

Answer

Ethical considerations are irrelevant when using probability models in decision making.

Answer

Probability models should only be used in situations with complete certainty.

Question 33

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) = (1 / sqrt(2 * pi * sigma^2)) * exp(-(x - mu)^2 / (2 * sigma^2))

Answer

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

Answer

P(X = x) = (1 / (2*pi)) * (exp(-(x^2)/2))

Question 34

What is the expected value of a discrete random variable X with probability mass function P(X = x)?

Accepted Answer

E(X) = Σ(x * P(X = x))

Answer

E(X) = Σ((x - mu)^2 * P(X = x))

Answer

E(X) = ∫(-∞ to ∞) x * f(x) dx

Answer

E(X) = σ^2

Question 35

Which of the following distributions is commonly used to model the time between events?

Accepted Answer

Exponential distribution

Answer

Poisson distribution

Answer

Normal distribution

Answer

Binomial distribution

Question 36

Which of the following is NOT a type of skewness?

Accepted Answer

Bimodal

Answer

Positive

Answer

Negative

Answer

Zero

Question 37

What is the purpose of Bayes' theorem?

Accepted Answer

To calculate the probability of an event based on prior knowledge and new evidence

Answer

To calculate the expected value of a random variable

Answer

To calculate the variance of a random variable

Answer

To calculate the standard deviation of a random variable

Question 38

Which of the following is a random variable?

Accepted Answer

The number of heads in a coin toss

Answer

The age of a person

Answer

The color of a car

Answer

The speed of light

Question 39

Define probability

Accepted Answer

The measure of the likelihood of an event occurring

Answer

The number of times an event occurs

Answer

The certainty of an event occurring

Question 40

Which of the following is an example of a discrete probability distribution?

Accepted Answer

Binomial distribution

Answer

Exponential distribution

Answer

Uniform distribution

Answer

Normal distribution

Question 41

Which of the following is a characteristic of a Poisson distribution?

Accepted Answer

The mean and variance are equal

Answer

The probability of an event occurring is constant

Answer

The distribution is always symmetric

Question 42

Which of the following is a measure of the central tendency of a probability distribution?

Accepted Answer

Mean

Answer

Range

Answer

Variance

Answer

Standard deviation

Question 43

Which of the following is NOT a valid probability distribution?

Accepted Answer

Bernoulli distribution with p > 1

Answer

Uniform distribution from 0 to 1

Answer

Binomial distribution with n trials and p success probability

Answer

Poisson distribution with λ as the mean

Question 44

Which of the following is the correct definition of conditional probability?

Accepted Answer

The probability of event A occurring, given that event B has already occurred.

Answer

The probability of two events occurring independently.

Answer

The complement of the probability of an event occurring.

Question 45

Which of the following is the expected value of a continuous random variable?

Accepted Answer

The integral of the random variable multiplied by its probability density function over its support.

Answer

The mean of the random variable.

Answer

The median of the random variable.

Question 46

Which of the following is the variance of a normal distribution?

Accepted Answer

σ²

Answer

μ²

Answer

σ

Answer

μ

Question 47

Which of the following is a consequence of the Central Limit Theorem?

Accepted Answer

The distribution of sample means of sufficiently large samples from any population will be approximately normal, regardless of the shape of the population distribution.

Answer

The sample mean will always be equal to the population mean.

Answer

The sample variance will always be equal to the population variance.

Question 48

Which of the following probability distributions is characterized by outcomes that take on a finite or countably infinite number of possible values?

Accepted Answer

Discrete probability distribution

Answer

Gaussian distribution

Answer

Continuous probability distribution

Answer

Exponential distribution

Question 49

Which theorem allows for the adjustment of probabilities based on new information?

Accepted Answer

Bayes' theorem

Answer

Central Limit theorem

Answer

Law of large numbers

Answer

Chebyshev's inequality

Question 50

What is the probability of selecting a black card from a standard deck of 52 cards, containing 26 black cards?

Accepted Answer

26/52

Answer

1/2

Answer

26/26

Answer

1/4

Question 51

Which of the following scenarios represents a compound event?

Accepted Answer

Rolling a six on a dice and then rolling an even number.

Answer

Flipping a coin and obtaining heads.

Answer

Measuring the temperature of a room.

Answer

Drawing a face card from a deck of cards.

Question 52

What is the probability of obtaining at least one head when flipping two fair coins?

Accepted Answer

3/4

Answer

1/8

Answer

1/4

Answer

1/2

Question 53

Which statistical measure is NOT considered a measure of central tendency?

Accepted Answer

Range

Answer

Mean

Answer

Mode

Answer

Median

Question 54

Which of the following is a property of the normal distribution?

Accepted Answer

It is symmetric about the mean.

Answer

The mean and median are always equal.

Answer

It has three inflection points.

Answer

It is right-skewed.

Question 55

Bayes' theorem is used to:

Accepted Answer

Calculate the probability of an event based on condition probability and prior information.

Answer

Determine the probability of an event occurring twice.

Answer

Calculate the expected value of a random variable.

Question 56

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If two balls are drawn at random with replacement, what is the probability that the first ball drawn is red and the second ball drawn is blue?

Accepted Answer

3/20

Answer

2/5

Answer

1/5

Answer

1/10

Question 57

Which of the following is NOT a type of statistical inference?

Accepted Answer

Data presentation

Answer

Hypothesis testing

Answer

Parameter estimation

Answer

Confidence interval estimation

Question 58

Which of the following is a non-parametric statistical test that is used to compare the medians of two independent samples?

Accepted Answer

Mann-Whitney U test

Answer

t-test

Answer

ANOVA

Answer

Chi-square test

Question 59

Two events, A and B, are independent if:

Accepted Answer

The occurrence of event A has no effect on the probability of occurrence of event B

Answer

The probability of event A is equal to the probability of event B

Answer

The intersection of event A and event B is an empty set

Question 60

Which of the following is NOT a type of random variable?

Accepted Answer

Categorical variable with infinite categories

Answer

Discrete variable

Answer

Continuous variable

Answer

Ordinal variable

Question 61

The expected value of a random variable is a measure of:

Accepted Answer

Its central tendency

Answer

Its cumulative distribution function

Answer

Its variability

Answer

Its probability distribution

Question 62

Which of the following is a property of the normal distribution?

Accepted Answer

It is a bell-shaped, symmetric distribution

Answer

Its mean is always less than its median

Answer

It is always bimodal

Answer

It is a discrete distribution

Question 63

Bayes' Theorem is primarily used for:

Accepted Answer

Calculating conditional probabilities

Answer

Testing the independence of two events

Answer

Finding the mean and variance of a random variable

Question 64

Which of the following is a limitation of probability theory?

Accepted Answer

It cannot account for uncertainty due to incomplete or subjective information

Answer

It is inaccurate for large sample sizes

Answer

It is too complex to use in real-world scenarios