Binomial Distribution Formula | Step by Step Calculation

Formula to Calculate Binomial Distribution

Binomial Distribution Formula is used to calculate probability of getting x successes in the n trials of the binomial experiment which are independent and the probability is derived by combination between number of the trials and number of successes represented by nCx is multiplied by probability of the success raised to power of number of successes represented by px which is further multiplied by probability of the failure raised to power of difference between number of success and number of the trials represented by (1-p) n-x.

The probability of obtaining x successes in n independent trials of a binomial experiment is given by the following formula of binomial distribution:

P(X) = _nC_xp^x(1-p)^n-x

where p is the probability of success

In the above equation, _nC_xis used, which is nothing but combinations formula. The formula to calculate combinations is given as _nC_x= n! / x! (n-x)! where n represents the number of items (independent trials) and x represents the number of items being chosen at a time (successes).

In case n=1 in a binomial distribution, the distribution is known as Bernoulli distribution. The mean of a binomial distribution is np. The variance of the binomial distribution is np(1-p).

Calculation of the Binomial Distribution (Step by Step)

The calculation of binomial distribution can be derived by using the following four simple steps:

Step 1: Calculate the combination between the number of trials and the number of successes. The formula for _nC_xis where n! = n*(n-1)*(n-2) . . . *2*1. For a number n, the factorial of n can be written as, n! = n*(n-1)! For instance, 5! is 5*4*3*2*1
Step 2: Calculate the probability of success raised to the power of the number of successes that is p^x.
Step 3: Calculate the probability of failure raised to the power of the difference between the number of successes and the number of trials. The probability of failure is 1-p. Thus, this refers to obtaining (1-p)^n-x
Step 4: Find out the product of the results obtained in Step 1, Step 2 and Step 3.

Examples

You can download this Binomial Distribution Formula Excel Template here – Binomial Distribution Formula Excel Template

Example #1

The number of trials (n) is 10. The probability of success (p) is 0.5. Do the calculation of binomial distribution to calculate the probability of getting exactly 6 successes.

Solution:

Use the following data for the calculation of binomial distribution.

Calculation of binomial distribution can be done as follows,

P(x=6) = ₁₀C₆*(0.5)⁶(1-0.5)^10-6

= (10!/6!(10-6)!)*0.015625*(0.5)⁴

= 210*0.015625*0.0625

Probability of Getting Exactly 6 Successes will be-

P(x=6) = 0.2051

The probability of getting exactly 6 successes is 0.2051

Example #2

A manager of an insurance company goes through the data of insurance policies sold by insurance salesmen working under him. He finds that 80% of the people who purchase motor insurance are men. He wants to find out that if 8 motor insurance owners are randomly selected, what would be the probability that exactly 5 of them are men.

Solution: We first have to find out what are n, p, and x.

Calculation of binomial distribution can be done as follows,

P(x=5) = ₈C₅*(0.8)⁵(1-0.8)^8-5

= (8! /5! (8-5)! )*0.32768*(0.2)³

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= 56*0.32768*0.008

Probability of Exactly 5 Successes will be-

P(x=5) = 0.14680064

The probability of exactly 5 motor insurance owners being men is 0.14680064.

Example #3

Hospital management is excited about the introduction of a new drug for treating cancer patients as the chance of a person being successfully treated by it is very high. The probability of a patient being successfully treated by the drug is 0.8. The drug is given to 10 patients. Find the probability of 9 or more patients being successfully treated by it.

Solution: We first have to find out what is n, p, and x.

We have to find the probability of 9 or more patients being successfully treated by it. Thus, either 9 or 10 patients are successfully treated by it

x (number that you have to find probability for) = 9 or x = 10

We have to find P(9) and P(10)

Calculation of binomial distribution to find P(x=9) can be done as follows,

P(x=9) = ₁₀C₉*(0.8)⁹(1-0.8)^10-9

= (10! /9! (10-9)!)*0.134217728*(0.2)¹

= 10*0.134217728*0.2

Probability of 9 Patients will be-

P(x=9) = 0.2684

Calculation of binomial distribution to find P(x=10) can be done as follows,

P(x=10) = ₁₀C₁₀*(0.8)¹⁰(1-0.8)^10-10

= (10!/10! (10-10)!)*0.107374182*(0.2)⁰

= 1*0.107374182*1

The probability of 10 Patients will be-

P(x=10) = 0.1074

Therefore, P(x=9)+P(x=10) = 0.268 + 0.1074

= 0.3758

Thus, the probability of 9 or more patients being treated by the drug is 0.375809638.

Binomial Distribution Calculator

You can use the following binomial distribution calculator.

n
p
x
Binomial Distribution Formula =

Binomial Distribution Formula =	_nC_x * p^x * (1 -p)^n-x
	₀C₀ * 0⁰ * (1-0)^0-0 =	0

Relevance and Use

There are only two outcomes
The probability of each outcome remains constant from trial to trial
There are a fixed number of trials
Each trial is independent i.e. mutually exclusive of others
It provides us with the frequency distribution of the possible number of successful outcomes in a given number of trials where each of these given trials has the same probability of success.
Each trial in a binomial experiment can result in just two possible outcomes. Hence, the name is ‘binomial’. One of these outcomes is known as success and the other as a failure. For instance, people who are sick may respond to a treatment or not.
Similarly, when we toss a coin, we can have only two types of outcomes: heads or tails. The binomial distribution is a discrete distribution used in statistics, which is different from a continuous distribution.

An example of a binomial experiment is tossing a coin, say thrice. When we flip a coin, only 2 outcomes are possible – heads and tails. The probability of each outcome is 0.5. Since the coin is tossed thrice, the number of trials is fixed that is 3. The probability of each toss is not influenced by other tosses.

Binomial distribution finds its applications in social science statistics. It is used for developing models for dichotomous outcome variables where there are two outcomes. An example of this is whether Republicans or Democrats would win the election.

Binomial Distribution Formula in Excel (with excel template)

Saurabh learned about the binomial distribution equation in school. He wants to discuss the concept with his sister and have a bet with her. He thought that he would toss an unbiased coin 10 times. He wants to bet $100 on getting exactly 5 tails in 10 tosses. For the purpose of this bet, he wants to compute the probability of getting exactly 5 tails in 10 tosses.

Solution: We first have to find out what is n, p, and x.

There is an inbuilt formula for binomial distribution is Excel which is

It is BINOM.DIST(number of successes, trials, a probability of success, FALSE).

For this example of the binomial distribution would be:

=BINOM.DIST(B2, B3, B4, FALSE) where cell B2 represents the number of successes, cell B3 represents the number of trials and cell B4 represents the probability of success.

Therefore, the calculation of Binomial Distribution will be-

P(x=5) = 0.24609375

The probability of getting exactly 5 tails in 10 tosses is 0.24609375

Note: FALSE in the above formula denotes the probability mass function. It calculates the probability of there being exactly n successes from n independent trials. TRUE denotes the Cumulative Distribution Function. It calculates the probability of there being at most x successes from n independent trials.