Probability Theory: Master the Fundamentals of Probability Distributions

Question 1

A fair six-sided die is rolled. What is the probability of rolling a number greater than 3?

Accepted Answer

1/2

Answer

1/3

Answer

1/4

Answer

1/6

Question 2

Which of the following is a benefit of using probability to make predictions?

Accepted Answer

It can help us understand uncertainty.

Answer

It is always accurate.

Answer

It is easy to apply.

Answer

It can be used to predict the future.

Question 3

Which of the following is NOT a characteristic of probability?

Accepted Answer

Accurately predicts the outcome of a random event

Answer

Values range between 0 and 1

Answer

Sum of probabilities for all outcomes in a sample space equals 1

Answer

Allows us to calculate the likelihood of events based on related events

Question 4

Which of the following precisely defines a probability distribution?

Accepted Answer

A function that assigns a probability to each potential outcome in a sample space

Answer

The complete set of all possible outcomes in a sample space

Answer

A tabular or graphical representation of outcome probabilities

Answer

A mathematical representation predicting the outcome of a random event

Question 5

In a normal distribution, mean and standard deviation are denoted as:

Accepted Answer

μ and σ

Answer

σ and μ

Answer

μ and z

Answer

z and σ

Question 6

Which of the following is a conditional probability type?

Accepted Answer

Posterior probability

Answer

Prior probability

Answer

Joint probability

Answer

Marginal probability

Question 7

The binomial distribution is used to model:

Accepted Answer

Number of heads in a series of coin flips

Answer

Time taken to complete a certain task

Answer

Height of a randomly selected person

Answer

Average score on a test

Question 8

A fundamental assumption of the Central Limit Theorem is:

Accepted Answer

Sample size is sufficiently large

Answer

Underlying population distribution is normal

Answer

Sample mean is normally distributed

Answer

Sample standard deviation is known

Question 9

In a hypothesis test for a population proportion, which statistic is used to calculate the test statistic?

Accepted Answer

(Sample proportion - hypothesized proportion) / standard error

Answer

(Sample mean - hypothesized mean) / standard error

Answer

Z-score

Answer

Chi-square statistic

Question 10

Which of the following is not a characteristic of a valid probability distribution?

Accepted Answer

Mean value is always positive

Answer

Non-negativity

Answer

Sum to 1

Answer

Variance is non-negative

Question 11

A coin is flipped twice. What is the probability of getting heads on both flips?

Accepted Answer

1/4

Answer

1/2

Answer

1

Answer

0

Question 12

Which of the following is a conditional probability?

Accepted Answer

The probability of event A given that event B has already occurred

Answer

The probability of event A

Answer

The probability of events A and B occurring simultaneously

Answer

The probability of either event A or event B occurring

Question 13

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

Accepted Answer

Observational study

Answer

Random experiment

Answer

Natural experiment

Answer

Controlled experiment

Question 14

What happens to the precision of a probability estimate when you increase the sample size?

Accepted Answer

Precision increases.

Answer

Precision decreases.

Answer

Precision stays the same.

Question 15

A standard deck of 52 cards is shuffled. What is the probability of drawing a face card (jack, queen, or king)?

Accepted Answer

3/13

Answer

1/4

Answer

1/5

Answer

1/6

Question 16

A lottery has 100 tickets, of which 10 are winning tickets. If an individual purchases 5 tickets, what is the probability of winning at least one prize?

Accepted Answer

0.4013

Answer

0.5

Answer

0.25

Answer

0.1

Question 17

A survey found that the likelihood of a student passing a math test is 0.7. If a class has 30 students, what is the anticipated number of students who will pass the test?

Accepted Answer

21

Answer

15

Answer

25

Answer

30

Question 18

A coin is flipped 10 times. What is the probability of getting at least one tail?

Accepted Answer

1 - (1/2)^10

Answer

10 * (1/2)^10

Answer

10 * (1/2)

Answer

(1/2)^10

Question 19

A group of students is taking a multiple-choice test with 20 questions, each question having four options. If a student randomly guesses on each question, what is the probability that the student will answer exactly 10 questions correctly?

Accepted Answer

(1/4)^10 * (3/4)^10

Answer

(3/4)^20

Answer

(1/4)^10 + (3/4)^10

Answer

(1/4)^20

Question 20

Which of the following is **not** a property of probability?

Accepted Answer

It can exceed 1.

Answer

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

Answer

It must be between 0 and 1.

Question 21

What is the probability of drawing a king or a queen from a standard deck of 52 cards?

Accepted Answer

1/4

Answer

1/8

Answer

1/2

Answer

1/16

Question 22

Which of the following probability distributions is characterized by a symmetrical, bell-shaped curve?

Accepted Answer

Normal distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Exponential distribution

Question 23

In a binomial distribution with n trials and probability of success p, what is the mean?

Accepted Answer

np

Answer

n(1-p)

Answer

p(1-n)

Answer

p^2

Question 24

What is the Bayes' Theorem primarily used for?

Accepted Answer

Calculating conditional probabilities

Answer

Finding the mean of a random variable

Answer

Determining the probability of joint events

Question 25

What is the key difference between empirical and theoretical probability?

Accepted Answer

Empirical probability is based on observed data, while theoretical probability is based on probability models.

Answer

Empirical probability is more accurate than theoretical probability.

Question 26

Which of the following is a limitation of probability theory?

Accepted Answer

It cannot account for uncertainty or randomness completely.

Answer

It assumes that all outcomes are equally likely.

Answer

It is only applicable to complex events.

Question 27

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of randomly selecting a marble that is NOT blue?

Accepted Answer

7/10

Answer

2/5

Answer

1/2

Answer

3/10

Question 28

Two dice are rolled. What is the probability that the sum of the two dice is 7?

Accepted Answer

1/6

Answer

1/4

Answer

1/12

Answer

1/36

Question 29

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

Accepted Answer

3/8

Answer

1/4

Answer

1/8

Answer

1/2

Question 30

In a standard deck of 52 cards, what is the probability of drawing a King or a Queen?

Accepted Answer

2/13

Answer

4/13

Answer

1/13

Answer

1/26

Question 31

A student is selected at random from a class where 60% of the students are female and 40% are male. If 70% of the female students and 50% of the male students have brown hair, what is the probability that a randomly selected student has brown hair?

Accepted Answer

62%

Answer

42%

Answer

55%

Answer

60%

Question 32

A random variable X has the following probability distribution: P(X=0) = 0.2, P(X=1) = 0.5, P(X=2) = 0.3. What is the expected value of X?

Accepted Answer

1.1

Answer

1

Answer

1.5

Answer

0.8

Question 33

Which of the following is NOT a property of a normal distribution?

Accepted Answer

The distribution is skewed to the left or right.

Answer

The distribution is symmetrical about the mean.

Answer

The mean, median, and mode are all equal.

Answer

The total area under the curve is equal to 1.

Question 34

The probability that a machine produces a defective item is 0.05. What is the probability that in a sample of 10 items, exactly 2 are defective?

Accepted Answer

0.0746

Answer

0.0105

Answer

0.914

Answer

0.25

Question 35

A company wants to estimate the average height of its employees. They randomly select 100 employees and measure their heights. What is the parameter of interest in this study?

Accepted Answer

The average height of all employees in the company

Answer

The average height of the 100 employees selected

Answer

The standard deviation of the heights of the 100 employees selected

Answer

The number of employees in the company

Question 36

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

Accepted Answer

The number of heads obtained when a coin is tossed 5 times

Answer

The temperature of a room

Answer

The height of a student

Question 37

A bag contains 5 red balls and 3 blue balls. If one ball is drawn at random, what is the probability that it is red?

Accepted Answer

5/8

Answer

3/8

Answer

5/3

Answer

3/5

Question 38

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

Accepted Answer

1/6

Answer

1/12

Answer

1/4

Answer

1/36

Question 39

A student is chosen at random from a class of 30 students. There are 18 girls in the class. What is the probability that the chosen student is a boy?

Accepted Answer

2/5

Answer

2/3

Answer

1/2

Answer

3/5

Question 40

In a standard deck of 52 cards, what is the probability of drawing a heart or a spade?

Accepted Answer

1/2

Answer

1/52

Answer

1/4

Answer

1/13

Question 41

What is the probability of getting heads at least once when flipping a fair coin twice?

Accepted Answer

3/4

Answer

1/2

Answer

1/4

Answer

1

Question 42

A box contains 4 green marbles, 5 red marbles, and 6 blue marbles. If two marbles are drawn one after the other without replacement, what is the probability that both marbles are red?

Accepted Answer

2/21

Answer

1/9

Answer

25/225

Answer

5/15

Question 43

The probability of an event occurring is 0.3. What is the probability of the event not occurring?

Accepted Answer

0.7

Answer

0

Answer

1

Answer

0.3

Question 44

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

Accepted Answer

The sum of all probabilities can be any value between 0 and 1

Answer

The sum of all probabilities must equal 1

Answer

The probability of each outcome is between 0 and 1

Question 45

A fair coin is flipped 100 times. According to the law of large numbers, what is the expected number of heads?

Accepted Answer

Approximately 50

Answer

Approximately 25

Answer

It's impossible to predict

Answer

Exactly 50

Question 46

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

Accepted Answer

Binomial distribution

Answer

Normal distribution

Answer

Exponential distribution

Answer

Uniform distribution (continuous)