Master Discrete Probability Distributions: Binomial, Poisson, and Geometric

Question 1

A standard die is rolled 10 times. What is the probability of rolling exactly four sixes?

Accepted Answer

0.2059

Answer

0.1029

Answer

0.0515

Question 2

The number of phone calls received per hour at a call center follows a Poisson distribution with a mean of 5. What is the probability of receiving exactly 3 calls in an hour?

Accepted Answer

(e^-5 * 5^3) / 3!

Answer

(5 * e^-5) / 3!

Answer

1 - e^-5

Answer

e^-3 * 5^3

Question 3

The time between arrivals of customers at a bank is geometrically distributed with a mean time of 10 minutes. What is the probability that the next customer will arrive within 5 minutes?

Accepted Answer

0.1

Answer

0.01

Answer

0.5

Answer

10

Question 4

The expected value of a discrete probability distribution is the average of all possible values, weighted by their probabilities.

Accepted Answer

True

Answer

Sometimes True

Answer

Not Applicable

Answer

False

Question 5

In a Poisson distribution, the expected value and variance are both equal to the mean, which is denoted by λ.

Accepted Answer

True

Answer

Only the variance is the mean

Answer

Only the expected value is the mean

Answer

False

Question 6

In a geometric distribution, the expected value is 1/p and the variance is 1/p^2, where p is the probability of success on each trial.

Accepted Answer

True

Answer

Only the expected value is 1/p

Answer

False

Answer

Only the variance is 1/p

Question 7

Which of the following is NOT a characteristic of a discrete probability distribution?

Accepted Answer

Values can be any real number

Answer

Finite or countable number of values

Answer

Probabilities sum to 1

Answer

Values can take on only certain outcomes

Question 8

The number of customers arriving at a store per minute follows a Poisson distribution with an average rate of 2 customers per minute. What is the expected number of customers arriving in a 5-minute period?

Accepted Answer

10

Answer

5

Answer

20

Answer

15

Question 9

A company has a success rate of 60% in winning contracts. If they submit 5 proposals, what is the probability that they will win exactly 3 contracts?

Accepted Answer

0.302

Answer

0.5

Answer

0.2

Answer

0.4

Question 10

The number of phone calls received by a call center per hour follows a Poisson distribution with an average of 5 calls per hour. What is the probability of receiving exactly 3 calls in a given hour?

Accepted Answer

0.1404

Answer

0.2327

Answer

0.0498

Question 11

The expected value of a binomial distribution with parameters n = 5 and p = 0.4 is:

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 12

The variance of a Poisson distribution with parameter λ = 3 is:

Accepted Answer

3

Answer

1

Answer

2

Question 13

A fair die is rolled twice. What is the probability of getting a sum of 7?

Accepted Answer

1/6

Answer

1/24

Answer

1/18

Answer

1/12

Question 14

The probability of winning a game is 0.6. If the game is played 10 times, what is the probability of winning at least 6 times?

Accepted Answer

0.7031

Answer

0.5000

Answer

0.3000

Answer

0.2969

Question 15

A factory produces light bulbs, with an average of 2 defective bulbs per 100. What is the probability of finding exactly 3 defective bulbs in a sample of 50?

Accepted Answer

0.0185

Answer

0.0093

Answer

0.0370

Question 16

What is the variance of a binomial distribution with n trials and probability of success p?

Accepted Answer

np(1-p)

Answer

np

Answer

p(1-p)

Answer

n(1-p)

Question 17

A machine produces widgets with a 5% defect rate. What is the probability that the first defective widget is found on the 10th attempt?

Accepted Answer

0.0405

Answer

0.005

Answer

0.95

Answer

0.05

Question 18

Which of the following is NOT a characteristic of a discrete probability distribution?

Accepted Answer

The probability of any outcome can be negative.

Answer

The random variable can only take on a finite number of values.

Answer

The probability of each outcome must be between 0 and 1.

Answer

The sum of all probabilities must equal 1.

Question 19

A company receives an average of 5 complaints per day. What is the probability of receiving exactly 3 complaints in a day?

Accepted Answer

0.1404

Answer

0.0313

Answer

0.2650

Answer

0.0842

Question 20

A basketball player has a 60% free throw success rate. What is the probability that he will miss his first 3 free throws and then make his fourth?

Accepted Answer

0.0384

Answer

0.6

Answer

0.1296

Answer

0.216

Question 21

Which of the following is NOT a characteristic of a discrete probability distribution?

Accepted Answer

The probability of each outcome can be negative.

Answer

The random variable can only take on a finite number of values.

Answer

The sum of all probabilities must equal 1.

Question 22

A company produces light bulbs with a mean lifespan of 1000 hours. What is the probability that a randomly selected light bulb will last less than 800 hours if the lifespan follows a Poisson distribution?

Accepted Answer

This cannot be calculated with the given information.

Answer

0.3679

Answer

0.1353

Question 23

What is the variance of a Poisson distribution with a mean of λ?

Accepted Answer

λ

Answer

√λ

Answer

λ^2

Answer

1/λ

Question 24

Which of the following distributions is most suitable for modelling the number of customers arriving at a store in a given time interval, assuming the arrivals are independent and occur at a constant rate?

Accepted Answer

Poisson

Answer

Normal

Answer

Binomial

Answer

Geometric

Question 25

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

Accepted Answer

1/6

Answer

1/12

Answer

1/3

Answer

1/4

Question 26

The number of phone calls received by a call center per minute follows a Poisson distribution with a mean of 3. What is the probability that the call center receives exactly 5 calls in a given minute?

Accepted Answer

(e^-3 * 3^5) / 5!

Answer

(e^-3 * 3^6) / 6!

Answer

(e^-3 * 3^3) / 3!

Answer

(e^-3 * 3^4) / 4!

Question 27

A basketball player has a 60% chance of making a free throw. What is the probability that the player makes exactly 3 free throws out of 5 attempts?

Accepted Answer

(5 choose 3) * (0.6)^3 * (0.4)^2

Answer

(5 choose 3) * (0.6)^3 * (0.4)^3

Answer

(5 choose 3) * (0.6)^2 * (0.4)^3

Answer

(5 choose 3) * (0.6)^4 * (0.4)^1

Question 28

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If two marbles are drawn from the bag without replacement, what is the probability that both marbles are blue?

Accepted Answer

9/70

Answer

12/70

Answer

3/10

Answer

6/70

Question 29

The probability mass function of a random variable X is given by P(X = x) = (1/3)^x * (2/3)^(3-x) for x = 1, 2, 3. What is the expected value of X?

Accepted Answer

2

Answer

3

Answer

1

Answer

2.5

Question 30

The probability generating function (pgf) of a geometric distribution is given by G(s) = ps / (1 - (1-p)s), where p is the probability of success. What is the expected value of a geometric distribution with parameter p?

Accepted Answer

1/p

Answer

1-p

Answer

p

Answer

p^2

Question 31

A discrete probability distribution has the following pmf: P(X = 0) = 0.2, P(X = 1) = 0.3, P(X = 2) = 0.4, and P(X = 3) = 0.1. What is the mode of this distribution?

Accepted Answer

2

Answer

3

Answer

1

Answer

0

Question 32

The median of a Poisson distribution with parameter λ is the largest integer n such that the cumulative distribution function (cdf) P(X <= n) is greater than or equal to 0.5. What is the median of a Poisson distribution with λ = 3?

Accepted Answer

3

Answer

4

Answer

2

Answer

5

Question 33

Which of the following is NOT a discrete probability distribution?

Accepted Answer

Normal distribution

Answer

Geometric distribution

Answer

Poisson distribution

Answer

Binomial distribution

Question 34

What is the probability of getting exactly 3 heads when flipping a coin 5 times?

Accepted Answer

0.3125

Answer

0.75

Answer

0.25

Answer

0.5

Question 35

The average number of goals scored by a team in a soccer match is 2. What is the probability that they score at least 3 goals in their next match?

Accepted Answer

0.3679

Answer

0.6321

Answer

0.2707

Answer

0.4801

Question 36

A company produces light bulbs with a probability of 0.05 of being defective. What is the probability that the 5th bulb produced is the first defective bulb?

Accepted Answer

0.2441

Answer

0.951

Answer

0.49

Answer

0.049

Question 37

Which of the following is true about the expected value of a discrete probability distribution?

Accepted Answer

It is the sum of the products of each possible outcome and its corresponding probability.

Answer

It is always a whole number.

Answer

It is always equal to the mean of the distribution.

Question 38

What is the variance of a binomial distribution with parameters n = 10 and p = 0.5?

Accepted Answer

2.5

Answer

10

Answer

5

Answer

0.5

Question 39

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

Accepted Answer

The mean and variance are equal.

Answer

The probability of success is constant for all values of x.

Answer

It is a continuous distribution.

Answer

It is used to model the number of goals scored in a soccer match.

Question 40

What is the mean of a geometric distribution with parameter p = 0.4?

Accepted Answer

2.5

Answer

0.04

Answer

0.4

Answer

10

Question 41

Which of the following is an application of discrete probability distributions?

Accepted Answer

Modeling the number of defective items produced by a factory.

Answer

Predicting the weather

Answer

Estimating the population of a country

Question 42

A fair dice is rolled twice. What is the probability of getting a sum of 7?

Accepted Answer

1/6

Answer

1/36

Answer

1/18

Answer

1/12

Question 43

Which of the following is NOT true about binomial distribution?

Accepted Answer

It can take any real value.

Answer

Its variance is np(1-p).

Answer

Its mean is np.

Answer

It is a discrete distribution.

Question 44

A discrete random variable X follows a Poisson distribution with parameter λ. What is the probability of X taking the value 2?

Accepted Answer

λ^2 * e^(-λ) / 2!

Answer

λ^3 * e^(-λ) / 3!

Answer

λ^3 * e^(-λ) / 2!

Answer

λ^2 * e^(-λ) / 1!

Question 45

Which of the following describes the geometric distribution?

Accepted Answer

Number of trials until the first success.

Answer

Number of trials between two consecutive successes.

Answer

Number of successes in a fixed number of trials.

Answer

Time until the next event occurs.

Question 46

A fair coin is tossed 4 times. What is the probability of getting exactly 2 heads?

Accepted Answer

6/16

Answer

8/16

Answer

4/16

Answer

10/16

Question 47

The number of phone calls received by a call center per hour follows a Poisson distribution with a mean of 5. What is the probability of receiving more than 8 calls in an hour?

Accepted Answer

0.1125

Answer

0.3679

Answer

0.2251

Answer

0.0563

Question 48

What is the expected value of a geometric distribution with parameter p?

Accepted Answer

1/p

Answer

1-p

Answer

p

Question 49

A multiple-choice test has 10 questions, each with 4 options. If a student guesses randomly on each question, what is the probability that they get exactly 5 questions correct?

Accepted Answer

252/256

Answer

63/256

Answer

126/256

Answer

31/256

Question 50

The probability of a radioactive atom decaying in the next second is 0.001. What is the probability that it decays within the next 5 seconds?

Accepted Answer

0.00499

Answer

0.00999

Answer

0.00749

Answer

0.00249

Question 51

Which of the following is the mean of a binomial distribution with n=10 and p=0.2?

Accepted Answer

2

Answer

3

Answer

1

Answer

4

Question 52

The number of customers arriving at a bank per minute follows a Poisson distribution with a mean of 3. What is the probability that more than 5 customers arrive in the next 2 minutes?

Accepted Answer

0.1038

Answer

0.1557

Answer

0.2076

Answer

0.0519