Binomial Distribution Calculator

A binomial distribution calculator helps determine the probability of achieving a specific number of successes in a sequence of events. Similar to this, a binomial probability calculator allows you to easily compute the binomial cumulative distribution function (CDF) and probability mass, given the probability of success on a single trial, the number of trials, and the number of events. It can calculate the probability of success if the outcome follows a binomial distribution, such as flipping a coin. Additionally, the calculator can determine the number of trials required. This blog shall explore what the binomial probability distribution is, when and how to apply it, and will help you learn the binomial probability formula, among other details. Read more.

What is a Binomial Distribution?

The binomial distribution X∼Bin(n,p) is a probability distribution that represents the number of events in a series of n independent experiments. Each experiment has a binary or Boolean outcome, such as true/false, yes/no, success/failure, or event/no event. These experiments are called Bernoulli trials, and a binomial distribution results from a sequence of such trials. 

The distribution is defined by two parameters: n, the number of independent experiments, and p, the probability of success in a single experiment. The binomial distribution is widely used in teaching probability and statistics because of its desirable properties, such as having a known standard deviation and easily computable cumulative distribution and inverse functions.

What is a Binomial Probability?

The probability of a specific outcome in a binomial distribution is often called a "binomial probability." This probability can be calculated using the binomial probability distribution function (PDF), also known as the probability mass function (PMF), denoted as f(x), with the following formula:

fx = Pr [X = x] = (n/x​) px  (1-p)n-x = n!/x! (n-x!) px (1-p)n-x , where

• X represents a random variable

• x is a specific outcome

• n is the number of trials, and 

• p is the probability of success on each trial.

• The term (n/x​) also called "n choose x," is the binomial coefficient, which indicates the number of ways to choose x unordered combinations from a set of n. This is essentially the number of possible combinations.

Please Note: In some variations of the formula, (1−p) is replaced by q, where q=1−p.

Binomial Cumulative Distribution Function

The cumulative distribution function (CDF) of the binomial distribution is used to calculate the probability of observing fewer or more than a specific number of events, outcomes, or successes across several trials. The formula for the Binomial CDF is given below:

                                x

F (x) = Pr [X ≤ x] = ∑  f (i)

                                i=0

The cumulative binomial probability is the sum of the probabilities for all outcomes from 0 up to x. Our binomial distribution calculator applies this formula to compute various cumulative probabilities, such as the probability of events being less than or equal to x, less than x, greater than or equal to x, and greater than x. All of these calculations are considered cumulative binomial probabilities.

Binomial Distribution Formula in Probability

In probability theory and statistics, the binomial distribution is a discrete probability distribution that accounts for experiments with only two possible outcomes: either Success or Failure. For example, when we toss a coin, there are two possible results: heads or tails. Similarly, in a test, the outcomes can be either pass or fail. This distribution is often referred to as the binomial probability distribution. Let us look at the binomial distribution formula in probability for any random variable X:

P(x:n,p) = nCx px (1-p)n-x

Or

P(x:n,p) = nCx px (q)n-x

Where,

• n = the number of experiments

• x = 0, 1, 2, 3, 4, …

• p = Probability of Success in a single experiment

• q = Probability of Failure in a single experiment = 1 – p

How to use the Binomial Distribution Calculator?

To use the binomial distribution calculator or binomial probability calculator, follow these simple steps:

• Enter the total number of independent trials (experiments) you are conducting. For example, if you are tossing a coin 10 times, enter 10.

• Enter the probability of success on a single trial. This value should be between 0 and 1. For instance, if the probability of getting heads in a coin toss is 0.5, enter 0.5.

• Enter the specific number of successes you want to calculate the probability for. For example, if you want to find the probability of getting exactly 6 heads in 10-coin tosses, enter 6.

• Press the Calculate button to get the result. The calculator will use the binomial distribution formula to compute the probability of exactly x successes in n trials with a probability p of success on each trial.

Wrapping Up

A binomial distribution calculator and a binomial probability calculator are essential tools for calculating the likelihood of specific outcomes in a series of experiments with binary results. These calculators simplify the process and provide accurate results. You can apply these concepts effectively in various fields, such as statistics, probability theory, and real-world scenarios like coin flips or pass/fail tests. The binomial distribution is a fundamental tool in probability, offering a simple approach to analysing and predicting outcomes based on binary events.

FAQs

What is a binomial distribution?

The binomial distribution X∼Bin(n,p) models the number of events in n independent experiments, each with a binary outcome (e.g., success/failure). These experiments are characterised by two parameters: 

• n = the number of trials) and

• p = the probability of success in each trial

What is a binomial experiment?

A binomial experiment involves a set number of independent trials, each with only two possible outcomes. These outcomes are usually classified as either success or failure.

What is a binomial probability?

Binomial probability refers to the likelihood of achieving exactly x successes in n trials of an experiment that has only two possible outcomes (either Success or Failure).

Is the binomial distribution discrete or continuous?

The binomial distribution is discrete. It is a generally used probability distribution.

How to calculate a binomial distribution?

You can calculate the binomial distribution formula in probability for any random variable X, using the formula:

P(x:n,p) = nCx px (1-p)n-x

Or

P(x:n,p) = nCx px (q)n-x

How to calculate binomial on calculator?

To calculate binomial on a calculator, you can use the binomial probability distribution function, given by the following formula:

fx = Pr [X = x] = (n/x​) px  (1-p)n-x = n!/x! (n-x!) px (1-p)n-x

How to solve nCr in binomial distribution?

To solve nCr in a binomial distribution, you can use the formula:

nCr=n!/ ((r!*(n-r)!) , where:   

• n = The total number of objects &

• r = The number of objects to be chosen       

How to calculate binomial probability?

You can calculate binomial probability by using the binomial probability distribution function (PDF), given as: 

fx = Pr [X = x] = (n/x​) px  (1-p)n-x = n!/x! (n-x!) px (1-p)n-x  

What is the cumulative distribution function?

The cumulative distribution function (CDF) calculates the probability that the random variable X is less than or equal to a specific value, x, and is commonly represented as F(x).