# Probability: Binomial, Normal & Poisson

Probability

A certain state is contemplating creating a weekly lottery, the revenues from which will be used to fund improvements in the state's public education system. The commission chartered to develop the guidelines for the proposed lottery envisions using a process whereby six (6) balls will be randomly selected from a single bin containing a total of forty (40) balls, each of which is individually numbered 1 through 40, in order to determine the winning lottery numbers each week. Once a given ball has been randomly selected it will not be placed back in the bin before selecting the next ball. The commission is currently debating whether an individual should be required to pick the winning lottery numbers in a specific order or be allowed to pick them in a random order.

1. Assuming that the order in which the winning lottery numbers are selected is irrelevant (i.e. not important), what is probability of someone correctly selecting the six (6) winning lottery numbers?

2. Assuming that the order in which the winning lottery numbers are selected is relevant (i.e. is important), what is probability of someone correctly selecting the six (6) winning lottery numbers?

33 students in a college Physics 101 course recently took a mid-term exam. 8 students earned an A, 8 students earned a B, 6 students earned a C, 5 students earned a D and 6 students earned an F on the exam. The students were queried regarding the number of hours they had devoted to studying for the exam. 7 of the students who earned an A, 6 of the students who earned a B, 4 of the students who earned a C, 1 of the students who earned a D, and 1 of the students who earned an F reported that they had devoted more than 8 hours to studying for the exam. The remaining students reported that they had devoted no more than 8 hours to studying for the exam.

3. What is the probability of a randomly selected student having earned an F on the exam?

4. What is the probability of a randomly selected student having devoted more than 8 hours to studying for the exam?

5. What is the probability of a randomly selected student having earned an F on the exam given they devoted no more than 8 hours to studying for the exam?

6. What is the probability of a randomly selected student having earned an A or a B on the exam given they devoted more than 8 hours to studying for the exam?

Frequency Distributions

A graduate student is conducting a study involving individuals who claim to have paranormal capabilities. Participants are asked to reach into a bag that contains seven green balls and three red balls and select a red ball using only their paranormal ability to determine which balls are red (i.e., the participants cannot see the balls in the bag when making their selection). Each participant is given 50 attempts to select a correct colored ball. An attempt is coded as a "success" when the participant selects a red ball or a "failure" when the participant selects a green ball (i.e., the study is structured as a binomial experiment).

7. What is the probability that a randomly selected participant will select exactly 15 red balls?

8. What is the probability that a randomly selected participant will select no more than 15 red balls?

9. What is the probability that a randomly selected participant will select more than15 red balls?

The time required to complete a certain type of construction project is normally distributed with a mean of 60 weeks and a standard deviation of 4 weeks.

10. What is the probability of the project completing in no more than 56 weeks?

11. What is the probability of the project completing in more than 64 weeks?

Customers arrive at a supermarket check-out counter following a Poisson distribution with an average arrival rate of 5 customers per hour. Customers are checked out following an exponential distribution with an average service rate of 6 customers per hour.

12. What is the probability of exactly 5 customers arriving at the supermarket check-out counter in a given one hour period?

13. What is the probability the customer service time will be less than or equal to the expected average service time in a given one hour period?

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#### Solution Summary

The solution provides step by step method for the calculation of binomial, normal and Poisson probabilities. Formula for the calculation and Interpretations of the results are also included.

12 Multiple Choice Word Problems involving the Binomial, Normal, Poisson and Exponential Probability Distributions

Need some assistance with the following questions.

Please provide your answers and supporting documentation in excel format for the following questions.The answers in excel should match the options provided.

A student has an important exam coming up and is contemplating not studying for the exam in order to attend a party with his friends. The student must earn a minimum score of 70% on the exam in order to successfully maintain his desired GPA. Suppose the student knows in advance that the exam will consist of twenty multiple choice questions with four possible answers for each question. Answer questions 1-3 using the preceding information and modeling this situation as a binomial distribution.

1. What is the probability that the student will successfully earn exactly the required minimum score of 70% on the exam based solely upon randomly guessing the correct answer for each question?

o 2.57

o 2.57E-02

o 2.57E-05

o 2.57E-04

2. What is the probability that the student will earn less than the required minimum score of 70% on the exam based solely upon randomly guessing the correct answer for each question?

o 0.74673

o 0.85198

o 0.99997

o 0.23499

3. What is the probability that the student will successfully earn no less than the required minimum score of 70% on the exam based solely upon randomly guessing the correct answer for each question?

o 3.51E-04

o 2.95E-05

o 6.87E-06

o 1.27E-03

The mean time required to complete a certain type of construction project is 52 weeks with a standard deviation of 3 weeks. Answer questions 4-7 using the preceding information and modeling this situation as a normal distribution.

4. What is the probability of the completing the project in no more than 52 weeks?

o 0.25

o 0.50

o 0.75

o 0.05

5. What is the probability of the completing the project in more than 55 weeks?

o 0.2743

o 0.5091

o 0.7511

o 0.0546

6. What is the probability of completing the project between 56 weeks and 64 weeks?

o 0.2587

o 0.3334

o 0.5876

o 0.0911

7. What is the probability of completing the project within plus or minus one standard deviation of the mean?

o 0.951

o 0.852

o 0.759

o 0.683

Customers arrive at a supermarket check-out counter with an average arrival rate of 9 customers per hour. Answer questions 8-10 using the preceding information and modeling this situation as a Poisson distribution.

8. What is the probability of less than 5 customers arriving at the supermarket check-out counter in a given one hour period?

o 0.054

o 0.446

o 0.359

o 0.612

9. What is the probability of exactly 12 customers arriving at the supermarket check-out counter in a given one hour period?

o 0.262

o 0.044

o 0.073

o 0.189

10. What is the probability of no less than 12 customers arriving at the supermarket check-out counter in a given one hour period?

o 0.115

o 0.197

o 0.381

o 0.686

A local commuter bus service advertises that buses run every twelve minutes along a certain route. Answer questions 11and 12 using the preceding information and modeling this situation as an exponential distribution.

11. What is the probability of a bus picking up the passengers at a given bus stop in less than or equal to 12 minutes following their arrival at the bus stop?

o 0.519

o 0.632

o 0.466

o 0.772

12. What is the probability of a bus picking up the passengers at a given bus stop in more than 15 minutes following their arrival at the bus stop?

o 0.287

o 0.343

o 0.541

o 0.119