Binomial Probability & Binomial Experiments

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Binomial probability can be used to determine the likelihood of a certain outcome in an experiment where there are only two possible outcomes (success and failure). In this lesson, learn how to apply binomial probability to a variety of situations.

Binomial Experiments

If you flip a coin ten times, what is the probability that you will get heads exactly eight times? It doesn't seem very likely that you would get exactly eight heads in ten flips, but it's certainly not impossible. Binomial probability can help you determine exactly HOW likely it is to get ANY number of heads (or tails) in a coin flip experiment like this.

In order to use binomial probability to determine the likelihood that an event will occur, you first need to determine if the experiment is a binomial experiment or not. In order for an experiment to be considered a binomial experiment, each trial can only result in two possible outcomes. A coin flip is a great example of this, because every time you flip the coin, it has to land on either heads or tails. The prefix bi always means two, so think of this to remember that a binomial experiment is one in which there are two possible outcomes.

It's also important that the probability of success (represented by the symbol p) is the same for all the trials. For the coin flip, there is a 50% chance that it will land on heads every time you flip it, so the coin flip experiment meets this condition, too!

Finally, no matter how many trials are performed (n = the number of trials), each trial must not be affected by the outcome of any other trials. This is also true for the coin flip experiment, because each time you flip the coin, the outcome is not affected by the results of any of the previous trials.

Your coin flip experiment definitely meets all of these requirements, so now we know that it IS a binomial experiment.

Binomial Probability

To determine the probability of getting heads exactly eight times in ten coin flips, you first need to know how many different ways there are to get exactly eight heads. You can do this by finding the total number of combinations that will give you number of successes that you want.


combinations of coin flips


Next, determine the probability of success and the probability of failure. In this case, there is a 50% chance of success with each flip, so the probability of success is 0.5 and the probability of failure is 0.5. You can use these to calculate the probability of getting exactly eight heads in any one of the 45 possible combinations. Let's look at one of those possibilities to see how this works. One way to get eight heads is to have the first eight flips turn up heads and the last two turn up tails. What is the probability of this exact thing happening?


probability of one outcome


Since there are 45 different ways to get eight heads, you can find the probability of getting ANY one of these 45 different possibilities by multiplying the probability of getting one of these outcomes by the total number of combinations (45).


probability of 8 heads


So, the probability of getting exactly eight heads when you flip a coin ten times is 0.0439 or 4.39%.

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