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Finding Binomial Probabilities Using Formulas: Process & Examples

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  • 0:04 Binomial Probabilities
  • 0:36 Identifying Binomial…
  • 1:46 Finding a Binomial Probability
  • 5:30 Lesson Summary
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

You can find the probability of getting a certain number of successes when conducting a binomial experiment. In this lesson, you will learn how to find this information using the binomial probabilities formula.

Binomial Probabilities

Dakota is working on a research project with his friends for a government class. He and his friends have to call ten people in their town and ask if they voted in the past election. Dakota found previous research that says the likelihood of someone voting in an election in his town is 20%. What is the probability that Dakota and his friends will find five people that voted?

To solve this problem, you need to understand binomial experiments and probabilities. In this lesson, you will learn how to identify binomial probabilities and solve problems using the binomial formula.

Identifying Binomial Probabilities

First, let's discuss how you can identify a binomial experiment. A binomial experiment is an experiment that contains a fixed number of trials that results in only one of two outcomes: success or failure. For example, a person flipping a coin ten times to see how many heads appear in the coin flips would be a binomial experiment. There are some things to keep in mind when learning about binomial experiments:

  • First, the outcomes must be independent. This means that the outcome of one trial cannot have any influence on another. We can assume that as Dakota and his friends are making calls, the first person they call will not have any influence on the second person they call, and so forth.
  • Second, a binomial experiment must only have two possible outcomes. In this case, the two possible outcomes are either the person voted or they didn't.
  • Third, there are a fixed number of trials in a binomial experiment. In Dakota's experiment, there are ten people he will call - this is a fixed number that he and his friends have determined before the experiment begins.

Now that you understand binomial experiments, let's practice finding binomial probabilities using the binomial formula.

Finding a Binomial Probability


The binomial probability formula
The binomial probability formula


This is the binomial formula. Before we get into how to use this formula, let's review the information that we've gathered so far. We know that Dakota and his friends are calling ten people. We know that there is a 20% chance that the people they call will have voted in the last election. We also know that Dakota and his friends want to know the probability of five people out of the ten having voted in the last election. So, what can we do with these numbers?

First, we need to find the values of x, n, and P. The x represents the number of successes, the n represents the number of trials, and the P represents the probability of success on an individual trial. In our case, x represents the number of people who voted in the last election.

Dakota and his friends want to know if 5 people voted in the last election, therefore x = 5. The n represents the number of people Dakota and his friends will call, therefore n = 10. The P represents the probability of an individual trial, and each person is 20% likely to say yes, therefore P = .20. When we plug our numbers into the formula, it should look like this. I've color coded the numbers so you can see where each belongs in the binomial probability formula.


Binomial probability formula


Now, you may be wondering, 'What does the C stand for?' The C stands for combination. This means that that we are looking at the probability of a combination of 5 people saying they have voted in the last election. It doesn't matter if it is the first 5 people, the last 5 people, or a combination in between. In this case, order does not matter. In order to solve this equation, we will first need to find the value of C. The combination formula looks like this.


The combination formula
The combination formula


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