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Calculating Binomial Probability: Formula & Examples

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  • 0:04 Binomial Probability
  • 2:56 Non-Desirable Success
  • 3:57 Extending the Idea
  • 4:52 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

When we're calculating probabilities and there are only two possible outcomes, we can consider using the binomial probability. In this lesson we'll look at the conditions for using this probability, the formula, and some examples that clarify how to use it.

Binomial Probability

Let's say that your star basketball player has a 65% accuracy with her long range three-point shots. What's the probability that she'll be successful with 4 of her next 5 attempts?

Mathematically, this is answered by calculating a binomial probability, which is the probability of an exact number of successes on a number of repeated trials in an experiment that can have just two outcomes. Let's explain the binomial probability formula and do several examples to show how to use it.

There are some key ingredients for this type of probability. First, we must have a fixed number of trials (N). In the three-point shot example, N is equal to 5. Each of these trials must be independent trials. Making the current shot is independent of any other attempt. Then, we have to have two and only two possible outcomes. The basketball shot is successful or it's not. That's where the 'bi-' prefix in the word 'binomial' comes in.

We have to agree as to what is the success outcome. This is important terminology, though later we will look at an example where the 'success' outcome is not beneficial. We also have to ask for a certain number of successes (k) out of the total number of trials. In our example, k is equal to 4 successes. Finally, we need the probability of success (p). In our example, this was 65% which we will write as p = 0.65. The binomial probability formula is written as follows:


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We read this as the probability of k successes out of N trials given that the probability of one success is p. What is the q in this equation? This is the probability of failure (q). Since we have only two outcomes, the probability of success plus the probability of failure is equal to one. Thus, q is 1 - p. In our basketball example, the probability of failure is 1 - p = 1 - 0.65 = 0.35.

The term with the large parentheses is called the binomial coefficient, or the number of combinations of N take k. It is calculated in general as:


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For this example it's calculated as:


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The exclamation point means factorial, and it's calculated as N! = N(N - 1)(N - 2) … 1. Thus, 5! = 5(4)(3)(2)(1) = 120. Remember that 0! = 1.

Thus, the probability of our basketball player making 4 out of the next 5 three-point shots is:


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Which is very good indeed!

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