Binomial Distribution

The binomial distribution is a model that measures the probability of a particular event occuring within a fixed number of trials. It is a discrete probability distribution that is used for studying the occurrence of a desired outcome. The model determines the number of trials required to achieve the desired outcome.

The model is limited by certain assumptions; each trial is assumed to have only an either-or outcome. The model assumes that the result of one trial does not affect the outcome of the next trial. If the number of failures incurred to reach the desired outcome is accounted for, it is considered a negative binomial distribution.

• The binomial distribution is a common probability distribution. It is applied to scenarios where only two possible outcomes are possible.
• The model works as a discrete distribution (countable outcomes) and is the opposite of a continuous distribution.
• The binomial distribution in statistics works on three important elements, the number of trials, the frequency of successful trials, and the probability of successful trials.

The binomial distribution is a probability distribution model. It follows a discrete distribution pattern where the variables can only be integers—the range is finite. This probability distribution works in a set of parameters and follows several assumptions. Therefore. It presents a limited scope for application. It can only be applied to scenarios that result in an either-or outcomes.

The proof for the probability model was published in 1713—after the death of Swiss mathematician Jakob Bernoulli. The binomial distribution conditions paint a picture where the probable outcome is studied and analyzed to make future predictions. These predictions assist in crucial financial and business decisions.

The binomial distribution graph forms a bell shape on the axis, referring to the increase in probability followed by a steep decline.

The Binomial Distribution Formula is used to calculate the probability of getting ‘x’ successes in the 'n' number of independent trials.

The probability is a combination of the number of the trials and the number of successes, represented by '_nC_x.' _nC_x is then multiplied by the probability of the success raised to the power of the number of successes represented by 'p^x,' further multiplied by the probability of failure raised to the power of difference between the number of success and number of the trials—represented by (1-p)^n-x.

Thus, the probability of obtaining x successes in 'n' independent trials of a binomial experiment is given by the following formula:

P(X) = _nC_x p^x(1-p)^n-x

• Here 'P' is the probability of success.

• In the above equation, _nC_x is used, which is nothing but a combination formula. The formula to calculate combinations is given as follows:

_nC_x = n! / x! (n-x)! 

• Here 'n' represents the number of items (independent trials), and 'x' represents the number of items being chosen at a time (successes).

• If n=1, the distribution is known as the Bernoulli distribution. The mean of binomial trials is expressed as 'np'. And the variance of the binomial experiments is denoted by  'np(1-p).'

Let us look at some examples to understand the application of binomial distributions.

Example #1

Let us assume that a school comprises forty students. In the given school, Jane, Jenny, and Judy are friends. The three friends study every day—six times a week. On a given day, an English test is taken. There are only two possible outcomes for the test results—either a success or a failure. For such a scenario, the binomial model checks the probability of all three students passing the test on all the days.

Therefore,

• The number of trials is limited, n = 6.
• The number of exclusive outcomes = 2 (pass or fail)
• All three friends are independent.

Other scenarios where a binomial model can be applied are as follows:

1. Conducting a survey for positive and negative reviews regarding any product or service.
2. Only registering a yes or no response to a question, situation, or scenario.
3. Counting male and female employees in an organization.
4. Accounting for votes in an election between two parties.

Example #2

Now let us look at an example with calculation.

The number of trials (n) is 10. The probability of success (p) is 0.5. Calculate the probability of getting exactly six successes.

Solution:
Given:

Based on the given values, the probability is calculated as follows:

P(x=6) = 10C6*(0.5)6(1-0.5)10-6

P(x=6) = (10!/6!(10-6)!)*0.015625*(0.5)4

P(x=6) = 210*0.015625*0.0625

Thus the probability of getting exactly 6 Successes is as follows:

P(x=6) = 0.2051

Thus, the probability of getting exactly 6 successes is 0.2051

For a binomial distribution, variance is less than the mean. With the Poisson distribution, on the other hand, variance and mean are equal. In contrast, for a negative binomial distribution, the variance is greater than the mean.

The mean, variance, and standard deviation for a given number of successes are represented as follows:

• Mean, μ = np
• Variance in binomial experiments is denoted by σ² = npq.
• Standard Deviation σ= √(npq)

In this sequence, p denotes the probability of success, and q represents the probability of failure, where q = 1-p.

It is also known as the Pascal Distribution for random variables in a negative binomial experiment. The failures are denoted by 'r.' The negative distribution concept sheds light on the number of trials required to attain a fixed number of successes. The failure frequency is denoted by 'r.'

A simple example would be pulling an Ace card from a deck of 52 cards. Here, pulling an ace is considered a failed outcome, and pulling any other card on repetitive attempts is considered a successful outcome.

So, if an ace is pulled out in the 5^th attempt, the number of failures will be denoted by r = 5, and the probability distribution of the pulling out non-Ace cards would be considered a negative binomial distribution.

The properties are as follows:

• The model only works in conditions where there are two exclusive outcomes—yes or no, pass or fail, heads or tail, true or false, etc.
• The number of trials or frequency of repetition is fixed.
• The success and failure outcomes fluctuate with each trial. The probability of success and failure depends on the former.
• All the trials and outcomes are independent. One trial does not influence the outcome of the next trial.

The differences are as follows:

• The binomial probability model is discrete. In contrast, the normal distribution is continuous.
• The range of the binomial model is finite. In contrast, normal distributions have an infinite range.
• With the Binomial model, the scope for application is limited. Normal distributions, on the other hand, are used to calculate everyday parameters like model height, population, weight, marks, etc.
• The Binomial model only allows integer values. Normal distribution allows real numbers as well.

This article has been a guide to what is Binomial Distribution & definition. We discuss its mean, variance, properties, binomial distribution vs normal distribution & calculation examples. You can learn more about it from the following articles -

• Multinomial Distribution
• Sampling Distribution
• Exponential Distribution