Back To Course

GMAT Prep: Help and Review25 chapters | 288 lessons | 15 flashcard sets

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Log in here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Tracy Payne, Ph.D.*

Tracy earned her doctorate from Vanderbilt University and has taught mathematics from preschool through graduate level statistics.

You have a probability distribution to create, which one do you use? That depends. In this lesson, learn about binomial distributions, get examples and criteria for their use, and learn how to calculate the binomial distribution formula.

A **probability distribution** is a function or rule that assigns probabilities of occurrence to each possible outcome of a random event. Probability distributions give us a visual representation of all possible outcomes of some event and the likelihood of obtaining one outcome relative to the other possible outcomes.

A **binomial distribution** is a specific probability distribution. It is used to model the probability of obtaining one of two outcomes, a certain number of times (*k*), out of fixed number of trials (*N*) of a **discrete random event**.

A binomial distribution has only two outcomes: the expected outcome is called a success and any other outcome is a failure. The probability of a successful outcome is *p* and the probability of a failure is 1 - *p*.

A successful outcome doesn't mean that it's a favorable outcome, but just the outcome being counted. Let's say a discrete random event was the number of persons shot by firearms last year. We'd be looking for the probability of obtaining some number of victims out of the pool of shootings. Being shot is neither a favorable nor a successful outcome for the victim, yet it is the outcome we are counting for this discrete variable.

The binomial distribution is used to model the probabilities of occurrences when specific rules are met.

- Rule #1: There are only two
**mutually exclusive**outcomes for a discrete random variable (i.e., success or failure). - Rule #2: There is a fixed number of
**repeated trials**(i.e., successive tests with no outcome excluded). - Rule #3: Each trial is an
**independent event**(meaning the result of one trial doesn't affect the results of subsequent trials). - Rule #4: The probability of success for each trial is fixed (i.e., the probability of obtaining a successful outcome is the same for all trials).

Binomial distributions would be used to model situations where the successful outcome is exactly one value. Let's look at some examples.

*Given a couple has 5 children, what is the probability that exactly 3 will be boys?*

Possible outcomes: boy or girl

Fixed number of repeated independent trials: 5

Out of 5 trials, exactly 3 children are boys = success

Probability of success (0.5) + probability of failure (0.5) = 1

*Given 5 questions on a test, what is the probability of randomly guessing exactly 2 questions correctly?*

Possible outcomes: right or wrong

Fixed number of repeated independent trials: 5

Out of 5 trials, exactly 2 questions answered correctly = success

Probability of success (0.5) + probability of failure (0.5) = 1

*Given 10 rolls of a die, what is the probability that you will roll the number 1 exactly five times?*

Possible outcomes: roll a 1 or roll something other than 1 (i.e., 2, 3, 4, 5, or 6)

Fixed number of repeated independent trials: 10

Out of 10 trials, exactly five of the rolls land on 1 = success

Probability of success (1/6 = 0.17) + probability of failure (5/6 = 0.83) = 1

What about when the successful outcome is not exactly one outcome? Let's rewrite the first situation as this:

*Given a couple has 5 children, what is the probability that 3 or more will be boys?*

Possible outcomes: boy or girl

Fixed number of repeated independent trials: 5

Out of 5 trials, either 3, 4, or 5 are boys = success

Probability of success (0.5) + probability of failure (0.5) = 1.

When the successful outcome takes on more than one exact value, then we are looking for the probability of a **cumulative binomial distribution**. Cumulative binomial distributions are calculated differently than when successes can take on only one value. The binomial distribution formula applies to situations that do not include cumulative probabilities.

To calculate a binomial distribution, you will need to (a) plug the correct value into each variable, (b) find the binomial coefficient, and (c) evaluate the binomial probability formula. Here we go!

**Question:** Given a couple has 5 children, what is the probability that exactly 3 are boys?

(a) First, plug the correct value into each variable:

*n*= number of independent trials = 5*k*= success = 3*p*= probability of success = 0.5

(b) Next, find the **binomial coefficient**:

(c) Then, evaluate the **binomial probability formula**:

**Answer:** Given a couple has 5 children, the probability that exactly 3 of them are boys is 0.3125.

A **probability distribution** is a function or rule that assigns probabilities of occurrence to each possible outcome of a random event. A **binomial distribution** is one kind of probability distribution used to model the probability of obtaining one of two outcomes, a certain number of times (*k*), out of a fixed number of trials (*N*) of a discrete random event.

To use a binomial distribution, the situation being modeled must adhere to four criteria:

- There can only be two mutually exclusive outcomes for a discrete random variable.
- There must be a fixed number of repeated trials.
- Trials must be independent.
- The probability of success for each trial is fixed.

When the successful outcome can take on more than one exact value, then we are looking for the probability of a **cumulative binomial distribution**.

To calculate a binomial distribution, identify the number of independent trials, number of successful trials, and the probability of success, and then evaluate the binomial probability formula.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Log in here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
15 in chapter 4 of the course:

Back To Course

GMAT Prep: Help and Review25 chapters | 288 lessons | 15 flashcard sets

- Statistical Analysis with Categorical Data 5:20
- Understanding Bar Graphs and Pie Charts 9:36
- Summarizing Categorical Data using Tables 4:57
- How to Calculate Percent Increase with Relative & Cumulative Frequency Tables 5:47
- What is a Two-Way Table? 3:40
- Make Estimates and Predictions from Categorical Data 3:13
- What is Quantitative Data? - Definition & Examples 4:11
- What is a Histogram in Math? - Definition & Examples 5:12
- What are Center, Shape, and Spread? 6:11
- How to Calculate Mean, Median, Mode & Range 8:30
- Describing the Relationship between Two Quantitative Variables 4:44
- Reading and Interpreting Line Graphs 6:09
- Making Estimates and Predictions using Quantitative Data 4:07
- Z Test: Formula & Example 5:28
- Binomial Distribution: Definition, Formula & Examples 6:14
- Data Abstraction: Definition & Example 5:08
- Types of Statistical Analysis 7:09
- How to Draw a Trend Line
- How to Find the Equation of a Trend Line 6:31
- Go to Data & Statistics: Help and Review

- SIE Exam Study Guide
- Indiana Real Estate Broker Exam Study Guide
- Grammar & Sentence Structure Lesson Plans
- Foundations of Science Lesson Plans
- Career, Life, & Technical Skills Lesson Plans
- Business Costs, Taxes & Inventory Valuations
- Using Math for Financial Analysis
- Assessments in Health Education Programs
- Governmental Health Regulations
- Understanding Health Education Programs
- AFOQT Prep Product Comparison
- ACT Prep Product Comparison
- CGAP Prep Product Comparison
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison

- El Hombre que se Convirtio en Perro: Author, Summary & Theme
- Achilles in The Iliad: Character Analysis & Description
- A Wrinkle in Time Chapter 5 Summary
- Population Composition: Definition & Elements
- Industrialization in Japan: Origins, Characteristics & Impact
- Baker v. Carr: Summary, Decision & Significance
- What is Deadlock? - Definition, Examples & Avoidance
- Quiz & Worksheet - Shang Dynasty Religion & Culture
- Quiz & Worksheet - Alternative Assessment Types
- Quiz & Worksheet - Animal Farm's Benjamin
- Quiz & Worksheet - Minimalist Painters
- Analytical & Non-Euclidean Geometry Flashcards
- Flashcards - Measurement & Experimental Design
- 5th Grade Math Worksheets & Printables
- Algebra 2 Worksheets

- High School Precalculus Syllabus Resource & Lesson Plans
- The Civil War and Reconstruction: Help and Review
- Leadership Study Guide
- GACE School Counseling (603): Practice & Study Guide
- Holt McDougal Biology: Online Textbook Help
- NMTA: Civil Rights
- GACE Middle Grades Math: The Pythagorean Theorem
- Quiz & Worksheet - Effective Lesson Plans
- Quiz & Worksheet - Economic Life & Growth in Nigeria
- Quiz & Worksheet - Economic Development
- Quiz & Worksheet - Finding Perimeter, Area & Volume of Combined Figures
- Quiz & Worksheet - Influence of Religion on UK's Public Life

- Theoretical Analyses of Religion
- Alkaloid Isolation & Purification
- Sequencing Activities for First Grade
- How to Publish in a Scientific Journal
- Transportation Lesson Plan
- Common Core State Standards in Missouri
- Finding Travel Grants for Teachers
- Study.com's Top Online Business Management Training Courses
- Activities for Building Team Motivation
- Best GMAT Prep Course
- Place Value Lesson Plan
- Texas Teacher Retirement & Social Security

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject