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Statistics 101: Principles of Statistics11 chapters | 141 lessons | 9 flashcard sets

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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.

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.

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.

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.

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.

You may notice that this formula uses an exclamation point, also known as a factorial in mathematics. Probability and statistics problems don't often use factorials, except when it comes to combinations. You will need to use a graphing calculator, or try a search on the Internet for 10! to find the factorial values. For more information about factorials, check out our other lessons.

The combination for this probability is 252. This means that there are 252 different combinations for this problem. Let's insert our combination into our formula and solve our binomial probability.

Alright, here is the work for our binomial probability formula. On the third row, you'll notice I plugged in our combination value, and I subtracted 5 from 10 in the last exponent. In this next row, I subtract .20 from 1. This part of the formula actually makes a lot of sense. If you think about it logically, there is a 20% chance that a person will say yes to Dakota's question.

That's the probability of success, but what about the probability of failure? That's something else that must be taken into consideration. If there is a 20% chance of success, then there must also be an 80% chance of failure. If you add the two together, you get 100%!

Okay, so now let's look at the next row. Here, I've solved .20 to the 5th power, which is 0.00032. In the next row, I do the same thing for the probability of failure, which is 0.32768. Now I just need to multiply from left to right. First, when I multiply 252 by 0.00032 I get 0.08064, and when I multiply that by 0.32768 I get 0.0264241152, rounded to 3%, which leads us to our answer: there is a 3% chance that 5 people out of 10 will have voted in last year's election.

Remember that a **binomial experiment** is an experiment that contains a fixed number of trials that results in only one of two outcomes: success or failure. We can use this information to find the probability of certain numbers of success. We can do this by using the binomial probability formula, which looks like this.

You will also need to find the combination using the combination formula, which looks like this.

Remember, if you have any trouble solving these formulas, break it down and pay close attention to the order of operations. Check out our other lessons for practice problems on this formula!

View the video lesson, then ensure your ability to:

- List the properties of a binomial experiment
- State the variables needed to calculate binomial probability
- Write the binomial probability formula and the combination formula
- Calculate a binomial probability

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Statistics 101: Principles of Statistics11 chapters | 141 lessons | 9 flashcard sets

- Go to Probability

- Random Variables: Definition, Types & Examples 9:53
- Finding & Interpreting the Expected Value of a Discrete Random Variable 5:25
- Developing Discrete Probability Distributions Theoretically & Finding Expected Values 9:21
- Developing Discrete Probability Distributions Empirically & Finding Expected Values 10:09
- Dice: Finding Expected Values of Games of Chance 13:36
- Blackjack: Finding Expected Values of Games of Chance with Cards 8:41
- Poker: Finding Expected Values of High Hands 9:38
- Poker: Finding Expected Values of Low Hands 8:38
- Lotteries: Finding Expected Values of Games of Chance 11:58
- Comparing Game Strategies Using Expected Values: Process & Examples 4:31
- How to Apply Discrete Probability Concepts to Problem Solving 7:35
- Binomial Experiments: Definition, Characteristics & Examples 4:46
- Finding Binomial Probabilities Using Formulas: Process & Examples 6:10
- Finding Binomial Probabilities Using Tables 8:26
- Mean & Standard Deviation of a Binomial Random Variable: Formula & Example 6:34
- Solving Problems with Binomial Experiments: Steps & Example 5:03
- Go to Discrete Probability Distributions

- Go to Sampling

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