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Binomial Distribution: Definition, Formula & Examples

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  • 0:00 What Is a Binomial…
  • 1:15 Criteria for Using…
  • 1:55 When a Successful…
  • 3:22 When a Successful…
  • 4:13 Evaluating Binomial…
  • 5:08 Lesson Summary
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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.

What Is a Binomial Distribution?

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.

Criteria for Using Binomial Distributions

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).

When a Successful Outcome is an Equality

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

When a Successful Outcome is an Inequality

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

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