# Binomial Distribution Calculator

Binomial Distribution Calculator calculates the binomial probability distribution for the given values. Binomial Distribution is the most widely used type of discrete probability distribution.

## What is a Binomial Distribution Calculator?

Binomial Distribution Calculator is an online tool that helps to calculate the binomial probability for any binomial random variable. The number of trials and the probability of success are the two parameters used in this distribution. To use this * binomial distribution calculator*, enter the values in the input boxes.

### Binomial Distribution Calculator

## How to Use Binomial Distribution Calculator?

Please follow the steps below to find the binomial distribution probability using the online binomial distribution calculator:

**Step 1:**Go to Cuemath’s online binomial distribution calculator.**Step 2:**Enter the probability of success in a single trial, number of successes desired, and number of trials in the given input boxes.**Step 3:**Click on the**"Calculate"**button to find the binomial probability**Step 4:**Click on the**"Reset"**button to clear the fields and enter new values.

## How Does Binomial Distribution Calculator Work?

The binomial distribution formula is used to represent 'r' successes in an experiment that is conducted 'n' times when we know that the probability of success in one trial is given by 'p'. The Binomial distribution is applicable to events in which the experiment has only two possible outcomes. This means that the experiment can either be a success or a failure. For example, binomial distribution can be used to find the number of females and males in a classroom. Given below is the formula for a binomial distribution.

P (r : n,p) = \(\binom{n}{r}\)p^{r}^{ }(1-p)^{n-r}

This formula can also be written as

P (r : n,p) = \(\binom{n}{r}\)p^{r}^{ }q^{n-r}

Here,

\(\binom{n}{r}\) = ^{n} \(C_{r}\) = = n! / [r! (n-r)!]

n = the number of trials.

r = the number of successes that are desired.

p = the probability of getting a success in one trial.

q = the probability of getting a failure in one trial. Furthermore, q = 1 - p.

When the value of n = 1, then the Binomial Distribution is known as a Bernoulli Distribution.

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## Solved Examples on Binomial Distribution

**Example 1:** A coin is tossed 7 times. What is the probability of getting exactly 4 heads in these 7 tosses and verify it using the binomial distribution calculator.

**Solution:**

Given: Number of trials = 7 and Number of success = 4

Probability of getting heads in a single coin toss = 0.5

Now, probability of getting 4 heads in 7 tosses = b(r; n, p) = \(\binom{n}{r}\)p^{r}^{ }(1-p)^{n-r}

b(4; 7, 0.5) = \(\binom{7}{4}\) × (0.5)^{4} × (1 – 0.5)^{7 - 4}

b(4; 7, 0.5) = 0.2734

Therefore, the probability of getting exactly 4 heads in these 7 tosses is 0.2734.

**Example 2:** A study was conducted with 5 people. The research showed that the probability of a person living a healthy life after 30 years was 2/3. Find the probability that after 30 years exactly 2 people are living and verify it using the binomial distribution calculator.

**Solution:**

n = 5

r = 2

p = 2/3 = 0.67

b(r; n, p) = \(\binom{n}{r}\)p^{r}^{ }(1-p)^{n-r}

b(2; 5, 0.67) = \(\binom{5}{2}\) × (0.67)^{5} × (1 – 0.5)^{5}^{ - 2}

b(2; 5, 0.67) = 0.164

Thus, the probability of 2 people living after 30 years is 0.164

Similarly, you can use the binomial distribution calculator and find the binomial distribution for:

- A laboratory claims that 3 out of 100 people have negative side effects of a drug test. To check this another laboratory conducts the test on 5 people. What is the probability that exactly 3 people experience these negative effects?
- Number of trails = 7, probability of success of a single trial = 0.8, and number of successes = 3

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