Binomial Distribution

The binomial distribution is a discrete distribution displaying data that has only TWO OUTCOMES and each trial includes replacement.

Such as:

  • GO / NO-GO
  • IN / OUT
  • HOT / COLD
  • HIGH / LOW


  1. Each trial has only two outcomes
  2. The experiment has n identical trials
  3. Each trial is independent of the other trials
  4. The probability of getting one outcome (success) p is held constant and the probability of getting the other outcome (failure) is also held constant, represented by (1 - p).
  5. Includes replacement for each trial. It can be used to approximate without replacement trials but it is suggested to use hypergeometric distribution formulas.

    Let's evaluate the assumptions:

    A coin flip is an example of two outcomes and each flip has the same chance for both outcomes (meaning independent). Flipping the coin n amount of times means there are n identical and independent trials. Nothing changes from one flip to another and the odds are the same for each flip.
Binomial Distribution Central Tendencies

The following formula is used to compute the number of experimental outcomes resulting from x successes in n trials.

Binomial Experimental Outcomes

For example: 4! (4 factorial) = 4*3*2*1 = 24

The Binomial Distribution Probability Function is shown below:

Binomial Probability Function

Example One

A manufacturing process creates 3.4% defective parts. A sample of 10 parts from the production process is selected. What is the probability that the sample contains exactly 3 defective parts?

There are two outcomes: Defective / Not-Defective, therefore the Binomial Distribution equation is applied.

p = 0.034
n = 10
then q = 1-p = 1 - 0.034 = 0.966 (96.6%)
Need the probability that x = 3.

Substitute the values into the Binomial Probability Function and solve:

Example One of applying the Binomial Distribution

Example Two

Forty-five percent of all registered voters in a national election are female. A random sample of 8 voters is selected. The probability that the sample contains 2 males is:

55% are male and there are two outcomes: MALE / FEMALE
n = 8
p = 0.55 (55%)
q = 1-p = 0.45
Need the probability that x = 2

Example Two of applying the Binomial Distribution

Example Three

79% percent of the students of a large class passed the final exam. A random sample of 4 students are selected to be analyzed by the school. What is the probability that the sample contains fewer than 2 students that passed the exam?

There are two outcomes: PASS / FAIL

p = 0.79
n = 4

Solving for the probability that x < 2.

Note that we need to compute p(x<2). So, it is necessary to add the P(x=0) to the P(x=1).

Each P(x=0) and P(x=1) must be calculated separately and added.

p(x<2) = p(x=0) + p(x=1)

Substitute the following values into the Binomial equation to find p(x=0):

n = 4
p = 0.79
q = 1 - p = 1 - 0.79 = 0.21
x = 0


n = 4
p = 0.79
q = 1 - p = 1 - 0.79 = 0.21
x = 1

Example Three of applying the Binomial Distribution

There is 3.12% chance of selecting fewer than 2 students that passed the exam when randomly selecting 4 students when 79% of them passed.

There are also Binomial tables that can be used when the input variables are known (and found within the table).

Application of the Binomial Distribution


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Other Discrete Distributions

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