# Binomial Calculator: How to Calculate Binomials with Ease

Published: 4/26/2021 Last updated: 2/24/2023

Binomial calculator is a tool that can be used to solve various statistical problems related to binomial distributions. It allows you to calculate the probabilities of certain outcomes or events based on the number of trials and the probability of success or failure in each trial. In this article, we will discuss the definition of binomial distribution, formula, examples, and frequently asked questions about binomial calculator.

## Definition of Binomial Distribution

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. The probability of success in each trial is denoted by p, and the probability of failure is denoted by q, where q = 1 - p. The binomial distribution is denoted by B(n,p), where n is the number of trials and p is the probability of success in each trial.

## Formula for Binomial Distribution

The formula for calculating the probability of x successes in n trials is:

P(x) = (n choose x) * p^x * q^(n-x)

where (n choose x) is the binomial coefficient, which is calculated using the formula:

(n choose x) = n! / (x! * (n-x)!)

where ! denotes the factorial function.

## Examples of Binomial Calculator

Let's look at some examples of how the binomial calculator can be used to solve problems related to binomial distributions.

#### Example 1:

A coin is flipped 5 times. What is the probability of getting exactly 3 heads?

Solution: In this case, n = 5 and p = 0.5 (assuming the coin is fair). The probability of getting exactly 3 heads is:

P(x=3) = (5 choose 3) * 0.5^3 * 0.5^(5-3) = 10 * 0.125 * 0.125 = 0.3125

Therefore, the probability of getting exactly 3 heads is 0.3125 or 31.25%.

#### Example 2:

A basketball player makes 80% of his free throws. If he shoots 10 free throws, what is the probability that he will make at least 8 of them?

Solution: In this case, n = 10 and p = 0.8. The probability of making at least 8 free throws is:

P(x>=8) = P(x=8) + P(x=9) + P(x=10) = (10 choose 8) * 0.8^8 * 0.2^2 + (10 choose 9) * 0.8^9 * 0.2^1 + (10 choose 10) * 0.8^10 * 0.2^0 = 0.3758

Therefore, the probability of making at least 8 free throws is 0.3758 or 37.58%.

#### What is the difference between binomial and normal distribution?

The binomial distribution is used when the number of trials is fixed and the probability of success or failure is known. The normal distribution is used when the data is continuous and the distribution is bell-shaped. The binomial distribution is a discrete probability distribution, while the normal distribution is a continuous probability distribution.

#### What is the mean and standard deviation of binomial distribution?

The mean of binomial distribution is μ = np, where n is the number of trials and p is the probability of success in each trial. The standard deviation of binomial distribution is σ = sqrt(npq), where q = 1 - p.

#### Can binomial distribution be used for non-binary outcomes?

No, the binomial distribution is only used for binary outcomes where each trial has only two possible outcomes: success or failure. If the outcomes are not binary, then other probability distributions such as Poisson or normal distribution should be used.

#### Can binomial calculator be used for large values of n?

For large values of n and small values of p or q, the binomial distribution can be approximated by the normal distribution using the central limit theorem. However, for small values of n, the binomial calculator should be used instead.

## Conclusion

The binomial calculator is a useful tool for solving problems related to binomial distributions. It allows you to calculate the probabilities of certain outcomes or events based on the number of trials and the probability of success or failure in each trial. In this article, we discussed the definition of binomial distribution, formula, examples, and frequently asked questions about binomial calculator. By understanding the binomial distribution and using the binomial calculator, you can make better statistical decisions in various fields such as finance, marketing, and sports.

## References

1. "Binomial Distribution." Stat Trek. https://stattrek.com/probability-distributions/binomial.aspx

2. "Binomial Distribution Calculator." Calculator Soup. https://www.calculatorsoup.com/calculators/statistics/binomial-distribution-calculator.php

3. "Binomial Distribution." Math is Fun. https://www.mathsisfun.com/data/binomial-distribution.html