Binomial Probability Distribution Formula

The binomial probability distribution formula only determines the chances of success (x) in the total n no. of trails. It can implement the binomial probability formula in binomial experiments independent of each other.

However, before digging deep into the [Binomial probability formula], there are a few important terms that you should be aware of:

Random variable:

The random variable is the variable where It can include the random values in an experiment or trial. For example, in a company with two different departments, It picked ten employees randomly to find out how many women were working in each department.

The first department randomly picked ten employees, and four were female.

In the 2nd department , on randomly picking the ten employees, six employees were female.

Here the variable of no. of women in each group has random values, i.e., 4 and 6 is called random variable.

Binomial trial:

An experiment where the outcome classifies it from two results, i.e., either success or failure, is called a binomial trial. For example, a coin tosses ten times where the outcome as the head will be a success and the tail takes as a failure in each trial.

Condition for binomial distribution experiment

  1. There should be a fixed number of trials in an investigation.
  2. Every problem in the study should be independent of each other, i.e., the result of any 1st trial must not affect the outcome of 2nd or other experiment trials and vice versa.
  3. The probability of getting success for each shot should be constant.
  4. Every problem in the experiment should only return one of the two possible outcomes, i.e., either success or failure.

Binomial probability formula

The binomial probability formula is as follows;

P(X) = nCx px(1-p)n-x


P is the probability of success

N is the number of trials

X is the no. of successes

Implementing binomial distribution

In the toy manufacturing industry, 100 toys from a batch cycle pick them to find the total number of defective pieces. Here the probability of finding ■■■■■ toys is p, while the random variable X (success) is the total no. of times the lousy toy finds them from 100 samples. Here binomial distribution represents as B (30, p).


When an experiment or survey repeats numerous times, the likelihood that the results will be SUCCESS or FAILURE is known as the binomial distribution. The prefix “bi” implies “two” or “twice”; therefore, the term “binomial” refers to a distribution type with two probable outcomes.

Distribution criteria

Binomial distributions must also have the following three requirements:

1- The number of studies or observations is predetermined. In other words, unless you perform something a specific number of times, you cannot calculate the likelihood that some event will occur. Common sense dictates that if you flip a coin once, your chance of obtaining tails is 50%. The possibility of receiving bottoms after 20 flips of the coin is close to 100%.

2- Each trial or observation stands alone. To put it another way, none of your problems have any bearing on the likelihood of the subsequent trial.

3- From one attempt to the next, the chances of success (tails, heads, fail, or pass) are precisely the same.

Frequently asked questions

The following are the most asked questions about the binomial probability distribution formula.

1- What do you mean by binomial?

An algebraic equation with two non-zero terms is called a binomial. Binomial language examples: The nominal with two variables, a and b, is a2 + 2b. A combination of two variables, x and y, is 5x3 - 9y2.

2- What characteristics do binomials have?

An experiment that meets these criteria is called a binomial experiment.

(1) There are n identical trials in the experiment.

(2) One of the two outcomes—success S or failure F—occurs in each trial.

(3) From one trial to the next, the likelihood of success is constant and equal to p.

3- The name “binomial distribution” refers to what?

The likelihood of k such events in n iterations, as determined by Swiss mathematician Jakob Bernoulli, is proportional to the kth element (where k starts with 0) in the development of the binomial formula (p + q)n, where q = 1 p. (Therefore, the term “binomial distributions”).

4- What does a binomial surname look?

The method used to name organisms is called the binomial classification system. There is a two-part name assigned to each species. Genus belongs to the first element, and the species’ name is the second. The honeybee, Apis mellifera (the honey bee)

5- What function does the binomial distribution serve?

With the use of the binomial distribution model, we can determine the likelihood of witnessing a certain number of “successes” when the system is carried out multiple numbers of times (for example, in a group of patients), and the result for each patient can either be a success or a failure.


A split is an integer that can produce by evaluating a smooth quadratic with two terms in statistics, more notably in arithmetic. It is a Cunningham statistic that has been generalized.