Probability Theory Quizzes: Test Your Understanding

Question 1

Which of the following probability distributions is continuous?

Accepted Answer

Normal distribution

Answer

Binomial distribution

Answer

Negative binomial distribution

Answer

Poisson distribution

Question 2

Which of the following is NOT a type of probability mass function?

Accepted Answer

Continuous probability distribution

Answer

Joint probability mass function

Answer

Conditional probability mass function

Answer

Marginal probability mass function

Question 3

Given two independent events A and B, the probability of both occurring is:

Accepted Answer

P(A) * P(B)

Answer

P(A) + P(B)

Answer

P(A) - P(B)

Answer

P(B) / P(A)

Question 4

What is the sample space for a single roll of a fair six-sided die?

Accepted Answer

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

Answer

{1, 2, 3, 4, 5, 6, 7}

Answer

{0, 1, 2, 3, 4, 5, 6}

Answer

{0, 1, 2, 3, 4, 5}

Question 5

Which of the following represents a valid probability distribution for the random variable representing the number of heads obtained when a coin is flipped three times?

Accepted Answer

{0, 0.33, 0.66, 1}

Answer

{0, 1, 2, 3}

Answer

{0.1, 0.2, 0.3}

Answer

{0.25, 0.5, 0.75, 1}

Question 6

If X and Y are mutually exclusive events, which statement is true?

Accepted Answer

P(X or Y) = P(X) + P(Y)

Answer

P(X or Y) = P(X) - P(Y)

Answer

P(X or Y) = 0

Answer

P(X or Y) = P(X) + P(Y) + P(X and Y)

Question 7

A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?

Accepted Answer

1/3

Answer

1/2

Answer

1/4

Answer

1/6

Question 8

A random variable X follows a uniform distribution over the interval [0, 1]. What is the variance of X?

Accepted Answer

1/12

Answer

1/6

Answer

1/4

Answer

1/2

Question 9

A fair coin is tossed three times, and the random variable X represents the number of heads obtained. Is X a discrete or continuous random variable? Explain your reasoning.

Accepted Answer

Discrete; X can only take specific values (0, 1, 2, or 3).

Answer

Continuous; X can take any value within the range [0, 3].

Answer

Discrete; X can only take the values 0 or 1.

Answer

Continuous; X can take any value less than 3.

Question 10

A six-sided die is rolled. What is the probability of rolling an even number?

Accepted Answer

1/2

Answer

1/3

Answer

2/3

Answer

1/4

Question 11

Which of the following is a valid event in the sample space {1, 2, 3, 4, 5}?

Accepted Answer

The number rolled is even.

Answer

The number rolled is negative.

Answer

Both numbers rolled are odd.

Answer

The number rolled is greater than 7.

Question 12

Which of the following probability distributions models the number of successes in a sequence of independent Bernoulli trials?

Accepted Answer

Binomial distribution

Answer

Normal distribution

Answer

Poisson distribution

Answer

Exponential distribution

Question 13

Which of the following is a key assumption of the central limit theorem?

Accepted Answer

The distribution of the sample mean will be approximately normal for large enough sample sizes.

Answer

The sample mean is a good estimate of the population mean.

Answer

The population from which the sample is drawn must be normally distributed.

Answer

The sample size must be less than 30.

Question 14

What is the expected value of a random variable X that takes the values 1, 2, 3, and 4 with probabilities 0.2, 0.3, 0.4, and 0.1, respectively?

Accepted Answer

2.2

Answer

0.5

Answer

1.5

Answer

2.7

Question 15

When tossing a coin, the probability of getting heads is:

Accepted Answer

0.5

Answer

0

Answer

1

Answer

Cannot be determined

Question 16

The cumulative distribution function (CDF) of a random variable X gives:

Accepted Answer

The probability of X being less than or equal to a given value

Answer

The probability of X being greater than a given value

Answer

The expected value of X

Answer

The standard deviation of X

Question 17

A sample space is the set of all possible:

Accepted Answer

Outcomes of an experiment

Answer

Events in an experiment

Answer

Random variables in an experiment

Answer

Probability distributions in an experiment

Question 18

The probability of an event A, denoted as P(A), is computed as:

Accepted Answer

Number of favourable outcomes / Number of possible outcomes

Answer

Number of unfavourable outcomes / Number of possible outcomes

Answer

Number of outcomes in A / Number of outcomes in the sample space

Answer

Number of outcomes not in A / Number of outcomes in the sample space

Question 19

The expected value of a random variable X, denoted as E(X), is given by:

Accepted Answer

Sum of all possible values of X multiplied by their probabilities

Answer

Difference of all possible values of X multiplied by their probabilities

Answer

Product of all possible values of X multiplied by their probabilities

Answer

Quotient of all possible values of X multiplied by their probabilities

Question 20

Which of the following is **not** a measure of variability?

Accepted Answer

Mean

Answer

Variance

Answer

Standard deviation

Answer

Range

Question 21

The central limit theorem states that the sampling distribution of sample means approaches which distribution as the sample size increases?

Accepted Answer

Normal distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Uniform distribution

Question 22

Which of the following is an example of a conditional probability?

Accepted Answer

Probability of getting a head when flipping a coin, given that the previous flip was a tail

Answer

Probability of rolling a 6 on a dice

Answer

Probability of drawing a red card from a deck of cards

Answer

Probability of winning a lottery

Question 23

Which of the following is **not** an application of probability theory?

Accepted Answer

Predicting the outcome of sports events

Answer

Predicting weather patterns

Answer

Analysing medical data

Answer

Determining the reliability of a system

Question 24

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

Accepted Answer

The probability of an event can be greater than 1.

Answer

Probability values range from 0 to 1, inclusive.

Answer

The sum of probabilities of all possible outcomes in a sample space is 1.

Answer

Probability is additive for mutually exclusive events.

Question 25

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

Accepted Answer

Zeta distribution

Answer

Binomial distribution

Answer

Exponential distribution

Answer

Normal distribution

Question 26

Which of the following theorems relates the expected value of a sum of random variables to the expected values of the individual random variables?

Accepted Answer

Linearity of Expectation

Answer

Law of Large Numbers

Answer

Central Limit Theorem

Answer

Chebyshev's Inequality

Question 27

What is NOT a property of probability?

Accepted Answer

A probability can be negative

Answer

Probability values range between 0 and 1

Answer

The sum of the probability of an event and its complement is 1

Answer

The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes

Question 28

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If a ball is drawn at random, what is the probability that it is not green?

Accepted Answer

4/5

Answer

1/2

Answer

2/5

Answer

3/5

Question 29

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

Accepted Answer

Number of heads in 10 coin flips

Answer

Height of a person

Answer

Temperature in a room

Answer

Time taken to complete a task

Question 30

Regarding independence of events, which statement is accurate?

Accepted Answer

Two events are independent if their joint probability equals the product of their individual probabilities

Answer

Two events are independent if the occurrence of one affects the probability of the other

Answer

Two events are independent if the sample space is equally likely

Answer

All events are automatically independent

Question 31

Which of the following is a continuous random variable?

Accepted Answer

Time taken to complete a task

Answer

Number of students in a class

Answer

Number of defective items in a production line

Answer

Number of heads in 10 coin flips

Question 32

The independence of two events A and B means that:

Accepted Answer

The occurrence of one event has no impact on the probability of the other event occurring.

Answer

The occurrence of one event guarantees the occurrence of the other event.

Answer

The events are mutually exclusive.

Question 33

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

Accepted Answer

1 - (1/2)^3

Answer

1/4

Answer

1/2

Answer

1/8

Question 34

A bag contains 5 red balls, 3 blue balls, and 2 green balls. If two balls are drawn with replacement, what is the probability of drawing a red ball and then a blue ball?

Accepted Answer

15/56

Answer

1/2

Answer

1/5

Answer

1/10

Question 35

Which of the following statements about conditional probability is correct?

Accepted Answer

Conditional probability is the probability of an event occurring given that another event has already occurred.

Answer

Conditional probability is always greater than or equal to the unconditional probability.

Answer

Conditional probability is only applicable to independent events.

Question 36

**Which of the following is a fundamental assumption of the Central Limit Theorem?**

Accepted Answer

The sample size is sufficiently large

Answer

The standard deviation of the sample must be known

Answer

The population distribution must be normal

Answer

The mean of the sample must be equal to the mean of the population

Question 37

**Which of the following is a common application of probability theory in real-world situations?**

Accepted Answer

Risk assessment

Answer

Solving differential equations

Answer

Predicting future events with certainty

Answer

None of the above

Question 38

**A Bayesian approach to probability theory differs from a frequentist approach mainly in its:**

Accepted Answer

Use of prior knowledge and subjective probabilities

Answer

Mathematical complexity

Answer

Focus on long-run frequencies

Question 39

**Which of the following is the sample space for the experiment of tossing a fair coin twice?**

Accepted Answer

{HH, HT, TH, TT}

Answer

{H, T}

Answer

{0, 2, 4, 6}

Answer

{0, 1, 2, 3}

Question 40

**If two events A and B are independent, which of the following is true?**

Accepted Answer

P(A and B) = P(A) * P(B)

Answer

P(A given B) = P(B given A)

Answer

P(A or B) = P(A) + P(B) - P(A and B)

Question 41

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

Accepted Answer

Variance

Answer

Mode

Answer

Mean

Answer

Median

Question 42

Which of the following is an application of probability theory?

Accepted Answer

Risk assessment

Answer

Calculus

Answer

Algebra

Answer

Geometry

Question 43

Which of the following is NOT a valid probability value?

Accepted Answer

-1

Answer

0

Answer

1

Answer

0.5

Question 44

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

Accepted Answer

1/13

Answer

1/4

Answer

1/26

Answer

1/52

Question 45

Two fair dice are rolled. What is the probability of rolling a sum of 7?

Accepted Answer

1/6

Answer

1/18

Answer

1/12

Answer

1/9

Question 46

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

Accepted Answer

0.8413

Answer

0.1587

Answer

0.6915

Answer

0.9772

Question 47

Which of the following is a key property of independent events?

Accepted Answer

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

Answer

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

Answer

The probability of at least one event occurring is equal to the product of their individual probabilities.

Question 48

A bag contains 6 red balls and 4 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

1/5

Answer

1/10

Answer

1/2

Answer

2/5

Question 49

Which of the following is a valid interpretation of the Bayes' Theorem?

Accepted Answer

It allows us to update our beliefs about an event based on new evidence.

Answer

It states that the probability of an event occurring is inversely proportional to the cost of observing it.

Answer

It provides a formula for calculating the expected value of a random variable.

Question 50

In a fair coin toss experiment, what is the expected number of heads in 100 tosses?

Accepted Answer

50

Answer

100

Answer

75

Answer

25

Question 51

An urn contains 10 balls, numbered 1 to 10. If three balls are drawn with replacement, what is the probability of drawing all odd-numbered balls?

Accepted Answer

1/8

Answer

1/32

Answer

1/16