A Detailed Guide to Binomial Probability Distribution Formula

The binomial probability distribution formula is used to know the chances of receiving success (x) in the total n no. of trails. The binomial probability formula can be implemented in binomial experiments that are independent of each other.
binomial
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 type of variable where the random values can be included in an experiment or trial. For example in a company with 2 different departments, 10 employees were picked randomly to find out how many women are working in each department.

In the first department on randomly picking ten employees, 4 employees were female.

In the 2nd department on randomly picking the 10 employees, 6 employees were female.

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

Binomial trial:

In an experiment where the outcome can be classified from two results, i.e. either success or failure are called binomial trials. For example, a coin is tossed 10 times where the outcome as the head will be taken as success and tail are taken as a failure in each trial.

Condition for binomial distribution experiment

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

Binomial probability formula

To know the probability of getting x (success) in independent trials of the binomial experiment the binomial probability formula is used:
P(X) = nCx px(1-p)n-x
Here,
P is the probability of success
N is the number of trials
X is the no. of successes

Implementing binomial distribution in industry or production domain:

Say, in Toy manufacturing in industry 100 toys from a batch cycle are picked to find out the total no. of defective pieces. Here the probability of finding defective toys is p while the random variable X (success) is the total no. of time the defective toy is found from 100 samples. Here binomial distribution represents as B (30, p).