# 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?

#### 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.