The binomial random variable is simply the probability that a survey or experiment will succeed or fail multiple times. Binomials are a type distribution with two possible outcomes (the bi prefix means two or more). A coin toss can only produce two outcomes, heads or tails. Taking a test may have two outcomes: pass or fail.

  • The number of experiments that have been run is the first variable in the binomial equation, n.
  • The probability of a specific outcome is represented by the second variable, p.

Let’s say, for example, you want to know how likely it is that you will get a 1. If you roll a die 20x, your probability of getting a 1 on any throw is 1/6. If you roll twenty times, the binomial distribution is (n=20; p=1/6). SUCCESS would mean “roll a one” while FAILURE would indicate “roll any other.” If the outcome was “roll a two,” the binomial distribution would become (n=20; p=1/2). This is because the probability of you throwing an even number equals one-half.

This is a specific type of discrete randomly variable that counts the number of times a given event occurs over a set number of trials or tries. To make a variable a binomial randomly variable, you must meet ALL the following conditions:

  1. A fixed number of trials is required (a fixed sample size).
  2. Each trial will show the event of interest.
  3. Each trial has the same probability of an occurrence (or lack thereof).
  4. Trials can be conducted independently of each other.


The following criteria must be met for binomial distributions:

  1. You can only determine the likelihood that something will happen if it is done a certain amount of times. This is common sense: if you toss one coin, your chance of getting a tails are 50%. Your chance of getting a tails if you toss the coin 20 times is very close to 100%.
  2. Every observation or trial is distinct. This means that none of your trials has an effect on the likelihood of the next trial.
  3. The success probability (tails and heads, fail or pass), is the exact same for each trial.

Once you have determined that your distribution is binomial you can use the binomial distribution algorithm to calculate the probability of binomial randomly variable.

What is a Binomial Random Distribution and how does it work? The Bernoulli Distribution

The Bernoulli distribution is closely related to the binomial distribution. Washington State University states that if each Bernoulli trial is an independent trial, then there will be a Binomial Distribution. The Bernoulli distribution, on the other hand, is the Binomial distribution where n=1.

Bernoulli Distribution is a collection of Bernoulli trials. Each Bernoulli trial can have one outcome. It can be either success or failure. The probability of success in each trial is P(S), = p. The probability of failure is just 1 minus the probability of success: P(F) = 1 – p. (Remember that “1” is the total probability of an event occurring…probability is always between zero and 1). All Bernoulli trials can be considered independent of each other. The probability of success is the same regardless of whether you have information about other trials’ outcomes.

What is a Random Binomial Variable and how does it work? Real-Life Examples

In real life, there are many examples of binomial distributions. If a new drug is introduced to treat a disease, it either cures it (it’s successful), or it fails to cure it (it’s unsuccessful). You can either win money or lose money if you buy a lottery ticket. A binomial distribution can represent any success or failure that you can think of binomial randomly variable.