Introduction to Probability: Master the Fundamentals of Chance and Uncertainty

Question 1

Two independent events, A and B, have probabilities of 0.6 and 0.4, respectively. What's the probability of both events happening?

Accepted Answer

0.24

Answer

0.1

Answer

0.8

Answer

1.0

Question 2

What is the sample space for the experiment of flipping a fair coin twice?

Accepted Answer

{HH, HT, TH, TT}

Answer

{Head, Tail}

Answer

{1, 2, 3, 4}

Answer

{Even, Odd}

Question 3

What is the probability of rolling a sum of 7 or 11 when rolling two fair six-sided dice?

Accepted Answer

1/12

Answer

1/6

Answer

1/18

Answer

1/36

Question 4

Which of the following is the complement of the event "A"?

Accepted Answer

Not A

Answer

A

Answer

A or Not A

Answer

A and Not A

Question 5

Which of the following is a characteristic of independent events?

Accepted Answer

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

Answer

The probability of both events occurring is lower than the probability of either event occurring individually.

Answer

The probability of one event occurring is dependent on the occurrence of the other event.

Answer

Independent events cannot occur simultaneously.

Question 6

Which of the following is not a characteristic of dependent events?

Accepted Answer

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

Answer

The occurrence of one event affects the probability of the other event occurring.

Answer

The probability of both events occurring is always less than the probability of either event occurring individually.

Answer

Dependent events can occur simultaneously.

Question 7

Under what condition is the conditional probability of event A given event B equal to the probability of A?

Accepted Answer

When A and B are independent events.

Answer

When A and B are mutually exclusive events.

Answer

When A and B are dependent events.

Answer

When A is a subset of B.

Question 8

Which of the following is not an axiom of probability?

Accepted Answer

The probability of the union of two events is always greater than the probability of each event.

Answer

The probability of an event is non-negative.

Answer

The probability of the sure event is 1.

Answer

The probability of an empty set is 0.

Question 9

The correlation coefficient between two random variables X and Y is 0.7. What does this indicate about the relationship between X and Y?

Accepted Answer

X and Y are positively correlated.

Answer

X and Y are negatively correlated.

Answer

X and Y are uncorrelated.

Answer

The direction of the correlation cannot be determined from the given information.

Question 10

Which of the following is a non-probability sampling method?

Accepted Answer

Convenience sampling

Answer

Simple random sampling

Answer

Stratified sampling

Answer

Cluster sampling

Question 11

In probability theory, what is a random variable?

Accepted Answer

A variable that assigns a numerical value to each outcome in a sample space

Answer

A constant with a fixed value for all outcomes

Answer

A variable whose value is determined by a non-random process

Question 12

Which of the following is not an axiom of probability?

Accepted Answer

The probability of an impossible event is 1.

Answer

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

Answer

The probability of the sample space is 1.

Answer

The probability of the union of two mutually exclusive events is the sum of their probabilities.

Question 13

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

Accepted Answer

It can take on negative values.

Answer

It is non-negative for all values of the random variable.

Answer

The area under the curve is equal to 1.

Answer

It describes the probability distribution of a continuous random variable.

Question 14

Which of the following is a fundamental principle of probability theory?

Accepted Answer

The probability of any event lies between 0 and 1.

Answer

The probability of the sample space is equal to 0.

Answer

The probability of the union of two mutually exclusive events is less than the sum of their probabilities.

Answer

The probability of the complement of an event is greater than the probability of the event.

Question 15

A fair coin is flipped 4 times. What is the probability of obtaining exactly 2 heads?

Accepted Answer

1/16

Answer

1/4

Answer

1/8

Answer

1/32

Question 16

Which of the following is a measure of central tendency?

Accepted Answer

Mean

Answer

Range

Answer

Variance

Answer

Standard deviation

Question 17

A data set has a mean of 8 and a variance of 16. What is the standard deviation?

Accepted Answer

4

Answer

2

Answer

8

Answer

16

Question 18

Which of the following probability distributions is used to model the number of successes in a sequence of independent Bernoulli trials?

Accepted Answer

Binomial distribution

Answer

Poisson distribution

Answer

Exponential distribution

Answer

Normal distribution

Question 19

Define a binomial distribution and explain its practical applications.

Accepted Answer

A binomial distribution models the number of successes in a series of independent experiments with a constant probability of success.

Answer

An exponential distribution models the time until the first success in a series of independent experiments.

Answer

A binomial distribution models the number of successes in a series of dependent experiments where previous outcomes affect probabilities.

Answer

A discrete uniform distribution models the probability of a specific number when rolling a dice.

Question 20

Which of the following is an **essential** element of a sample space?

Accepted Answer

A set of all possible outcomes

Answer

An outcome

Answer

An event

Answer

The probability of an event

Question 21

In a fair coin toss, what is the probability of getting either heads or tails?

Accepted Answer

1/2

Answer

1/4

Answer

1

Answer

0

Question 22

Which of the following pairs of events is dependent?

Accepted Answer

Drawing two cards from a deck without replacement

Answer

Tossing a coin and drawing a card from a deck

Answer

Rolling a die and flipping a coin

Answer

Rolling a 6 on a die and rolling a 3 on another die

Question 23

What is the probability of rolling an even number on a standard six-sided die?

Accepted Answer

1/2

Answer

1/3

Answer

1/4

Answer

1/6

Question 24

Which of the following is a conditional probability?

Accepted Answer

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

Answer

The probability of two events occurring simultaneously

Answer

The probability of an event occurring

Answer

The probability of an event not occurring

Question 25

Which of the following probability distributions is used to model the **waiting time** between arrivals in a queue?

Accepted Answer

Poisson distribution

Answer

Binomial distribution

Answer

Normal distribution

Answer

Chi-square distribution

Question 26

What is the variance of a binomial distribution with n = 10 and p = 0.4?

Accepted Answer

2.4

Answer

4

Answer

6

Answer

8

Question 27

How does the expected value help us determine the average outcome of a random variable?

Accepted Answer

The expected value is the average of all possible outcomes, weighted by their probabilities.

Answer

The expected value is the outcome that occurs most often.

Answer

The expected value is the outcome with the highest probability.

Answer

The expected value is the outcome that is most likely to occur given a certain event.

Question 28

An event is a subset of the sample space. Which of the following is NOT an event?

Accepted Answer

The set of all integers greater than 10

Answer

The set of all even numbers

Answer

The set of all outcomes in which a die lands on an even number

Answer

The set of all prime numbers

Question 29

Which of the following represents the sample space for flipping a coin twice?

Accepted Answer

{HH, HT, TH, TT}

Answer

{H, T, HH, HT}

Answer

{H, H, H, H}

Answer

{0, 1, 2, 3}

Question 30

If the probability of event A is 0.6 and the probability of event B is 0.4, what is the probability of either A or B occurring?

Accepted Answer

0.8

Answer

0.4

Answer

0.2

Answer

1.0

Question 31

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If a marble is drawn randomly from the bag, what is the probability of drawing a red or green marble?

Accepted Answer

7/10

Answer

1/3

Answer

1/2

Answer

2/5

Question 32

Which of the following events is independent of event A?

Accepted Answer

Event B if P(B|A) = P(B)

Answer

Event B if P(A|B) = P(B)

Answer

Event B if P(B) = P(A) + P(B)

Answer

Event B if P(AB) = 0

Question 33

Which of the following statements about variance is true?

Accepted Answer

Variance measures the spread of the probability distribution around its mean.

Answer

Variance can be negative.

Answer

Variance is always less than the standard deviation.

Answer

Variance is the probability of the most likely outcome.

Question 34

Which of the following is NOT an axiom of probability?

Accepted Answer

The probability of an event can be negative.

Answer

The probability of an event is greater than or equal to 0.

Answer

The probability of the sample space is 1.

Answer

The sum of the probabilities of all possible events is 1.

Question 35

A bag contains 4 red marbles, 3 blue marbles, and 2 green marbles. If one marble is drawn at random, what is the probability of drawing a blue marble?

Accepted Answer

3/9

Answer

1/3

Answer

1/2

Answer

2/5

Question 36

A fair coin is flipped twice. What is the probability of getting tails on the first flip and heads on the second flip?

Accepted Answer

1/4

Answer

1/2

Answer

1/8

Answer

1/16

Question 37

Which of the following is NOT an application of probability?

Accepted Answer

Predicting the winners of a sporting event

Answer

Quality control in manufacturing

Answer

Assessing the risk of a disease

Answer

Predicting the weather

Question 38

Which of the following probability distributions is used to model the number of successes in a sequence of independent trials?

Accepted Answer

Binomial distribution

Answer

Poisson distribution

Answer

Exponential distribution

Answer

Normal distribution