A binomial probability table can look intimidating to use. However, it can make your life a lot easier when trying to figure out binomial probabilities. This lesson will teach you how to read those tables.
Understanding Binomial Probabilities
McKenzi works for a local grocery store. The grocery store is starting a new marketing campaign to promote their cheese sales. First, the grocery store wants to do a little market research. McKenzi is in charge of asking 10 customers each hour if they like eating cheese. McKenzi wonders the probability of getting 8 people to say they like cheese in the first hour, 2 in the second hour, and 4 in the third hour. To find the answer to this question, McKenzi will need to understand binomial probabilities and how to read binomial distribution tables.
A binomial distribution table is a table of commonly used probability distributions created by statisticians. You can find binomial distribution tables right here.
McKenzi will be able to use binomial distribution tables to answer her question because she’s conducting a binomial experiment, which 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 10 times to see how many heads appear in the coin flips would be a binomial experiment. McKenzi’s work can be described as a binomial experiment, because either the person likes cheese, or they don’t. She also has a fixed number of trials: 10 for each hour. In fact, each hour that McKenzi works can be viewed as a separate binomial experiment.
There are some things to keep in mind when understanding binomial experiments. First, the outcomes must be independent. This means that the outcome of one trial cannot have any influence on another. In McKenzi’s experiment, we can assume that whether or not one customer said they liked cheese won’t affect what the next person reports.
Second, a binomial experiment must only have two possible outcomes. In this case, the two possible outcomes are either the person likes cheese, or they don’t.
Third, there are a fixed number of trials in a binomial experiment. For McKenzi’s work, she has to ask 10 people each hour. So remember, we can look at each hour as a separate binomial experiment.
Now that you understand binomial experiments, let’s look at binomial distributions and how McKenzi’s question can be answered using binomial distribution tables.
Binomial Distributions ; Tables
Before McKenzi starts work, she wants to figure out the binomial distribution for her experiment. Binomial distribution is the probability of each success in a given binomial experiment. In other words, binomial distribution shows us the probability for each scenario depending on the number of trials in the experiment. You will see two separate probabilities in this lesson. The first probability refers to the possibility of success on a trial. For example, if McKenzi were to flip a coin, then she has a 50% probability of getting heads and a 50% probability of getting tails. If McKenzi wanted to know the probability of successfully getting 5 heads in a row, then we would be working with the second form of probability in this scenario. Let’s discuss this further in McKenzi’s experiment.
In McKenzi’s experiment, she has 10 trials because she will be asking 10 people that hour if they like cheese. She can use a binomial distribution table like this one to determine the probability for each scenario.
Before you can read a binomial distribution table, you will need to understand some of the variables.
In all binomial distribution tables, 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. This probability is the same as the 50% probability of getting heads on one flip of the coin.
In McKenzi’s experiment, x represents how many people like cheese, n represents the 10 people she will ask and P represents the probability that someone will like cheese.
We already know how many trials there are in McKenzi’s experiment: 10, so we can input 10 for n. Now, we need to find P and x. First let’s look at P, or the probability of success of an individual trial. McKenzi’s manager tells her that market research consultants say the average person is 60% likely to eat cheese. Therefore, we can assume that P=.60. Now that we know n and P for our distribution, McKenzi can use the table to answer her questions about probability.
On the top of our table you will see n=10, but this n value will change depending on the number of trials in your experiment. Since McKenzi’s experiment will have 10 trials, we located the table for 10 trials.
The first row will have the probability, or what P equals. Locate P=.60 for McKenzi’s experiment. In this table, .6 is located on the last row.
The first column will have the number of successes. Remember, McKenzi wants to know the probability for 8, 2, and 4 successes. Locate x=8 on the table. It’s the 9th row down from the top.
If you follow the probabilities from x=8, over to P=.6, you will see that there is a .954 probability, or approximately 95%. This means that even though there is a 60% probability that an individual person will like cheese, there is a different probability for 8 out of 10 people liking cheese in this experiment.
Can you find the other two probabilities? Pause the video here to find the answer.
What’s the probability that McKenzi will only have 2 people say they like cheese out of the 10 that she asks? Right: .012, or approximately 1%. Now you may be wondering why McKenzi would have a lower probability of having 2 people say they like cheese, versus a higher probability of having 8 people say they like cheese. This is because the probability distribution looks at 2 people saying yes – and only those 2 people. It doesn’t say, ‘At least 2 people will say yes, they like cheese.’ It looks at, ‘Only 2 people will say yes to liking cheese, no more.’ This is why the probability is so low for x=2 – it is likely that more than 2 people will like cheese.
This is known as cumulative binomial distribution, which just means that the probability is a combination of the possible successes that are less than or equal to the trial number.
What about 4 people? The answer you should have is .166, or approximately 17%. This means that it is more likely that 4 people will say yes, they like cheese, than only 2 people saying they like cheese.
Remember, a binomial experiment is an experiment that contains a fixed number of trials that results in only one of two outcomes: success or failure. When performing this kind of experiment, you can determine the probability of certain successes by looking at a binomial distribution table. Binomial distribution is the probability of each success in a given binomial experiment. A binomial distribution table gives the probability of certain success of the experiment according to three different variables: 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.
To find your probability, simply locate the trial number, or n, that matches your experiment, then find the probability, or P, you’ve already determined, then locate your successes, or x, on the first row of the table. Follow that across to the probability and you will find the binomial distribution for that particular trial and number of successes. Remember, when analyzing a binomial distribution table, it will show cumulative binomial distribution, meaning that the probability is a combination of the possible successes that are less than or equal to the trial number.
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Seize the opportunity to realize the following objectives after watching the lesson on binomial probabilities:
- Enumerate three things that are essential to understanding binomial probabilities
- Recognize the function of a binomial distribution table
- Explain binomial distribution
- Demonstrate how to find a probability