# UTPA STEM/CBI Courses/Probability and Statistics/Binomial Distribution (Dr. Yanev)

Course Title: Probability and Statistics I

Lecture Topic: Binomial Distribution

Instructor: Dr. Yanev

Institution: University of Texas - Pan American

## Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
• Find the probabilities of different events related to a binomial random variable
• Work with the binomial distribution table on the back of the text
• Communicate their findngs about binomial distribution

• Sub Objectives- The objectives will require that students be able to:
• Understand the assumptions in the Bernoulli scheme
• Figure out the relationship between the mean and the binomial parameters.

• Difficulties- Students may have difficulty:
• Recognizing the binomial nature of the problem.
• Identifying the parameters of the binomial distribution, i.e., p and n.

• Real-World Contexts- There is a number of ways students can use this material in the real-world, such as:
• Building simple models based on binomial distribution.

Model of Knowledge

• Concept Map
• Understand the concept of a random variable.
• Concept of expected value.
• Content Priorities
• Enduring Understanding
• Understand the binomial model.
• Important to Do and Know
• Exprasing probabilities of events in terms of random variables.
• Working with probability tables.
• Worth Being Familiar with
• Binomial distribution formula.

Assessment of Learning

• Formative Assessment
• In Class (groups)
• Homework (individual)
• Pre-lecture quiz after student has read the section from the book
• Summative Assessment
• In class quiz over lesson

## Legacy Cycle

OBJECTIVE

By the next class period students will be able to:

• Find the probabilities of different events related to a binomial random variable
• Work with the binomial distribution table on the back of the text
• Communicate their findngs about binomial distribution

THE CHALLENGE

The students will be presented with the following problem. Mudlark Airlines has a 15-seater commuter airplane that is used for short flights. Their data suggest that on average about 8% of the customers who buy tickets are no-shows. Wanting to avoid empty seats (they see this as missed opportunity to increase revenue), they decide to sell more than 15 tickets for each flight. Ticketed customers who can't be seated on the plane will be accommodated on another flight and will receive a certificate for a free flight at another time. You have been hired as a consultant to Medlark. Your job is to determine how many tickets to sell. Explain your solution completely and write your recommendation to the company.

GENERATE IDEAS

The students will be given a few minutes to individually write down their answers to the questions that have been presented. Students may respond with answers such as:

• We should not overbooked the flight.
• We can sell one or two extra tickets.

MULTIPLE PERSPECTIVES

The professor will lecture on binomial distribution so that students will gain a better understanding of the topic. Students will be asked, "What do we need to know in order to answer our challenge questions?" and given a few minutes to individually write down their responses. Examples include:

• We need to know the probability of "success".
• We need to know the number of "trials".
• We need to find the probabilities of certain events.

RESEARCH & REVISE

Students will be provided with the formula for the expected value as function of the parameters and a table of binomial probabilities. They will then form groups and be asked to begin restating the problem in mathematical terms.

Students will receive feedback on their progress and ideas as well as the correct process for calculating the desired probabilities. Students will be presented with a quiz at the beginning of the following class.

GO PUBLIC

Students will now be able to answer the questions presented at the beginning of the lesson and will be asked to turn in a brief explanation. Use simulation methods as well as the binomial table from the textbook to perform your analysis.

## Pre-Lesson Quiz

Multiple Choice.

1. Which assumption below is not part of the Bernoulli scheme?

a) independent trials

b) two possible outcomes: success and failure

c) fixed probability of a success

d) infinite number of trials

e) fixed probability of a failure

2. What is the interpretation of a binomial random variable?

a) number of trials till the first success

b) number of trials till the first failure

c) number of sucesses in n trials

d) number of success in infinitely many trials

e) none of the above

3. What is the relationship between the expected value ${\displaystyle \mu }$ and the two parameters ${\displaystyle p}$ and ${\displaystyle n}$ of a binomial random variable?

a) ${\displaystyle \mu =n/p}$

b) ${\displaystyle \mu =pn}$

c) ${\displaystyle \mu =p/n}$

d) ${\displaystyle \mu =n^{p}}$

e) ${\displaystyle n=p\mu }$

NOVICE

The more tickets we sell the biger the risk is to end up with passengers without seats.

PRACTITIONER

How can you set up a sampling scheme and produce simulations?

MASTER

How do your conclusion depend on the level of risk you are willing to take in terms of probability of having passengers without seats?