Formula to Calculate Binomial Distribution

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

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where p is the probability of success

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

In case n=1 is in a binomial distribution, the distribution is known as the 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)

Examples

Example #1

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

  • Calculate the combination between the number of trials and the number of successes. The formula for nCx is where n! = n(n-1)(n-2) . . . 21. For a number n, the factorial of n can be written as n! = n(n-1)! For instance, 5! is 5432*1 Calculate the probability of success raised to the power of the number of successes that are px. 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 Find out the product of the results obtained in Step 1, Step 2, and Step 3.

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

               = 2100.0156250.0625

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 determine 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

              = 560.327680.008

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 introducing 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. Thus, either 9 or 10 patients are successfully treated by it.

x (a number that you have to find a 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

               = 100.1342177280.2

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

                  = 10.1073741821

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 with the drug is 0.375809638.

Binomial Distribution Calculator

You can use the following binomial distribution calculator.

Relevance and Use

  • There are only two outcomesThe probability of each outcome remains constant from trial to trialThere are a fixed number of trials.Each trial is independent, i.e., mutually exclusive of others.It provides us with the frequency distributionFrequency DistributionFrequency distribution refers to the repetitiveness of a variable, i.e., the number of times a variable occurs in a data set. In excel, it is a function to tabulate or graphically represent the recurrence of a particular value in a group or at an interval.read more of the possible number of successful outcomes in a given number of trials where each 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, sick people may respond to a treatment or not.Similarly, when tossing a coin, we can have only two outcomes: heads or tails. The binomial distribution is a discrete distribution used in statisticsStatisticsStatistics is the science behind identifying, collecting, organizing and summarizing, analyzing, interpreting, and finally, presenting such data, either qualitative or quantitative, which helps make better and effective decisions with relevance.read more, 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 two 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. Other tosses do not influence the probability of each toss.

Binomial distribution finds its applications in social science statistics. It is used to develop models for dichotomous outcome variables with 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 ten times. He wants to bet $100 on getting five tails in 10 tosses. For this bet, he wants to compute the probability of getting exactly five tails in 10 tosses.

There is an inbuilt formula for binomial distribution in Excel, which is:

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

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 exactly n successes from n independent trials. TRUE denotes the cumulative distribution function. It calculates the probability of at most x successes from n independent trials.

This article has been a guide to the Binomial Distribution Formula. Here, we learn how to calculate the probability of X using binomial distribution in Excel with examples and a downloadable Excel template. You can learn more about Excel modeling from the following articles: –

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