Binomial Distribution Formula
Before discussing the binomial distribution formula, let's understand the words binomial and distribution. A distribution is simply a collection of data, or scores, on a variable. Usually, these scores are arranged in order from smallest to largest, and then they can be presented graphically. A probability distribution is a mathematical function that provides the probabilities of the occurrence of various possible outcomes in an experiment.
In the word Binomial the prefix, ‘Bi’ means two or twice. A binomial distribution is the probability of a trial with only two outcomes. The binomial distribution has two different outcomes which are, success and failure. In probability, the binomial distribution comes with two parameters 'n' and 'p'. The first variable, n, stands for the number of times the experiment runs, and the second variable, p, represents the probability of one specific outcome.
Binomial distributions must also meet the following criteria:
 The number of observations or trials is fixed.
 Each observation or trial is independent.
 The probability of success is exactly the same from one trial to another.
The binomial distribution is closely related to the Bernoulli distribution and its origin can be traced back to Bernoulli's trial.
What Is Binomial Distribution Formula?
The binomial distribution formula is used to obtain the probability of observing 'x' successes in 'n' trials, with the probability of success on a single trial denoted by 'p'. The binomial distribution assumes that p is same for all trials.
The binomial distribution formula is:
\(b\left( {x;n,P} \right) = {}^nC_x .P^x .\left( {1  P} \right)^{n  x}\)
Where,
 b = binomial probability
 x = total number of successes
 P = probability of success on an individual trial
 n = number of trials
Note: The binomial distribution formula can also be written as,
\(P\left( x \right) = \frac{{n!}}{{\left( {n  x} \right)!x!}}.\left( p \right)^x .\left( q \right)^{n  x}\)
because \(^n C_x = \frac{{n!}}{{\left( {n  x} \right)!x!}}\) and q is just the probability of failure.
Let us understand the binomial distribution formula using solved examples.
Solved Examples Using Binomial Distribution Formula

Example 1: 60% of people who purchase sports cars are men. Find the probability that exactly 7 are men if 10 sports car owners are randomly selected.
Solution:
Let’s Identify ‘n’ and ‘X’ from the problem.
The number of sports car owners are randomly selected is n = 10, and
The number to find the probability is X = 7.Given: p = 60%, or 0.6.
Therefore, the probability of failure is q = 1 – 0.6 = 0.4
Now, using the binomial distribution formula
\( P( x ) = \frac{{n!}}{{( {n  x} )!x!}}.( p )^x .( q)^{n  x} \\
= \frac{{10!}}{{( {10  7} )!7!}}.( {0.6} )^7 .t( {0.4})^{10  7} \\
= 120 \times 0.0279936 \times 0.064 \\
= 0.215 \)
Answer: The probability that exactly 7 are men is 0.215 or 21.5%. 
Example 2: A coin is tossed 5 times. What is the probability of getting exactly 3 heads?
Solution:
Using the binomial distribution formula:
b(x; n, P): \(^nC_x \times P_x \times (1 – P)^{n – x}\)
Here,
The number of trials, n = 5
The odds of success (“tossing a heads”) is 0.5
So 1  p = 0.5
And x = 3
Now,\( b( {x;n,P} ) = {}^nC_x .P^x .( {1  P} )^{n  x} \\
= {}^5C_3 .( {0.5} )^3 .( {1  0.5})^{5  3} \\
= \frac{{5!}}{{( {5  3} )!3!}}.( {0.5} )^3 .( {0.5} )^2 \\
= 10 \times 0.125 \times 0.25 \\
= 0.3125 \)Answer: The probability of getting exactly 3 heads is 0.3125 or 31.25%.