## Introduction to Binomial Distribution

Binomial distribution is a type of probability distribution that is used to describe the probability of a certain number of successes in a fixed number of trials. It is one of the most commonly used probability distributions in statistics and probability theory.

In this article, we will provide a basic definition and explanation of the binomial distribution, its characteristics, and the method to find binomial probability.

## Definition and Explanation

The binomial distribution is a type of probability distribution that describes the probability of a certain number of successes in a fixed number of trials(known as “n”). It assumes that there are only two possible outcomes for each trial: success or failure.

This distribution is known for being discrete and has a limited range, where the number of successes occurring in the trials is limited between 0 and n.

In simpler terms, the binomial distribution is used to calculate the probability of a certain number of events occurring in a series of independent trials, where each trial has only two possible outcomes: success or failure.

## Characteristics

## The binomial distribution has several defining characteristics that are essential to understanding how this distribution works:

– Bernoulli Trials: The trials in the binomial distribution are often called Bernoulli trials. These are experiments with only two possible outcomes, and each trial is independent of the other trials.

– Success/failure: The binomial distribution assumes that there are only two possible outcomes for each trial: success or failure. – Probability of success: The probability of success is constant throughout all trials, denoted by “p”.

– Independence: The success or failure of one trial does not affect the outcome of any other trial.

## Method to Find Binomial Probability

## Formula to Calculate Binomial Probability

The formula to calculate the probability of X successes in n trials, with a probability of success p, is as follows:

P(X) = (n choose X) x (p)^X x (1-p)^(n-X)

## Where:

– (n choose X) is the binomial coefficient and is calculated as [n!/(X!(n-X)!)]. – (p)^X is the probability of getting X successes in n trials.

– (1-p)^(n-X) is the probability of getting (n-X) failures in n trials. This formula allows us to calculate the probability of a certain number of successes in a fixed number of trials with a known probability of success.

## Parameters of Binomial Distribution

The binomial distribution has two main parameters: the mean and the variance.

Mean: The mean of the binomial distribution is denoted by and is calculated as n*p (where n is the number of trials and p is the probability of success).

Variance: The variance of the binomial distribution is denoted by ^2 and is calculated as n*p*(1-p). The standard deviation can be found by taking the square root of the variance.

## Shape of Binomial Distribution Curve

The shape of the binomial distribution curve depends on the values of n and p. When there is a large number of trials (n) with a small probability of success (p), the distribution curve is skewed to the right.

When there is a small number of trials (n) with a high probability of success (p), the distribution curve is skewed to the left. When n and p have moderate values, the curve is bell-shaped and symmetrical.

## Conclusion

In conclusion, the binomial distribution is an essential concept in probability theory and statistics. It is used to calculate the probability of a certain number of successes or failures occurring in a fixed number of trials with a known probability of success.

Understanding the binomial distribution’s defining characteristics, formula, and parameters is crucial in many real-life scenarios, from quality control in manufacturing to medical research and more. With this knowledge, you can accurately predict and model the outcomes of independent trials, leading to better decision-making in various fields.

## How to Calculate Binomial Probability – Examples

The binomial distribution has many real-world applications in fields such as medicine, manufacturing, finance, and more. In this section, we will provide you with examples that demonstrate how to calculate the probability using the binomial distribution formula.

We will also show you how to calculate various probabilities in different instances to better understand the distribution.

## Probability Calculation Examples

Example 1: A Biased Coin

Let us consider a scenario where a coin is biased with a probability of getting heads in each toss being 0.6. What is the probability that in 8 tosses, we get exactly 4 heads? Solution: To calculate the probability of getting exactly 4 heads in 8 tosses, we will use the binomial formula:

P(X = 4) = (8 choose 4) x (0.6)^4 x (0.4)^4

= 0.2787

Therefore, the probability of getting exactly 4 heads in 8 tosses with a 0.6 probability of getting heads in each toss is 0.2787.

Example 2: Number of Trials

Suppose a manufacturing company reports that 4% of their products are defective. If 50 products are sampled, what is the probability that none of them are defective?

Solution: To calculate the probability that none of the 50 products are defective, we will once again use the binomial formula:

P(X = 0) = (50 choose 0) x (0.04)^0 x (0.96)^50

= 0.1813

Therefore, the probability that none of the 50 sampled products are defective, given the 4% probability of getting a defective product, is 0.1813.

## Calculation of Probabilities in Different Instances

Example 1: P(X = 5)

A company produces light bulbs with 2% chance of manufacturing faulty bulbs. Suppose that 200 light bulbs are produced, what is the probability that exactly 5 of them are faulty?

Solution: To calculate the probability that exactly 5 of the 200 produced light bulbs are faulty, we will use the binomial formula again:

P(X = 5) = (200 choose 5) x (0.02)^5 x (0.98)^195

= 0.028

Therefore, the probability that exactly 5 of the 200 produced light bulbs are faulty is 0.028. Example 2: P(X) < 4

Consider a scenario where a car dealer sells a new model that has 10% chance of having a manufacturing defect.

If a quality control inspector randomly selects six cars from the lot, what is the probability that less than four of them have a manufacturing defect? Solution: To calculate the probability that less than four of the six inspected cars have a manufacturing defect, we will find the sum of P(X = 0), P(X = 1), P(X = 2), and P(X = 3) using the binomial distribution formula.

P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

[(6 choose 0) x (0.1)^0 x (0.9)^6] + [(6 choose 1) x (0.1)^1 x (0.9)^5] + [(6 choose 2) x (0.1)^2 x (0.9)^4] + [(6 choose 3) x (0.1)^3 x (0.9)^3]

= 0.987

Therefore, the probability that less than four out of six inspected cars have a manufacturing defect is 0.987. Example 3: P(X) > 4

Suppose that a researcher has conducted a study on the effectiveness of a new medication for the common cold.

Based on previous studies, it is known that 30% of the study population is likely to benefit from the medication. The study includes 100 people; what is the probability that more than 4 of them benefit from the medication?

Solution: To calculate the probability that more than four of the 100 people in the study benefit from the medication, we have to find the complement of P(X 4). P(X > 4) = 1 – P(X 4)

= 1 – [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)]

= 1 – [(100 choose 0) x (0.3)^0 x (0.7)^100 + (100 choose 1) x (0.3)^1 x (0.7)^99 + (100 choose 2) x (0.3)^2 x (0.7)^98 + (100 choose 3) x (0.3)^3 x (0.7)^97 + (100 choose 4) x (0.3)^4 x (0.7)^96]

= 0.2585

Therefore, the probability that more than four out of the 100 people in the medication study benefit from the medication is 0.2585.

## Mean and Variance of Binomial Distribution

The mean and variance are important parameters of the binomial distribution that tell us about the distribution’s central tendency and spread. The mean (denoted by ) of a binomial distribution is equal to n x p, where n is the number of trials and p is the probability of success.

For example, if a coin is flipped 25 times with a probability of getting heads being 0.5, then the mean is 25 x 0.5 = 12.5.

The variance (denoted by ^2) of a binomial distribution is equal to n x p x (1 – p). For example, if in the same scenario above, where a coin is flipped 25 times with a probability of getting heads being 0.5, then the variance is 25 x 0.5 x (1 – 0.5) = 6.25.

Taking the square root of the variance gives us the standard deviation (denoted by ), which in this case is 2.5.

## Conclusion

The binomial distribution formula allows us to calculate the probability of a certain number of successes or failures in a fixed number of independent trials. By using examples and demonstrating how to calculate this probability in different instances, we can better grasp the concept’s practicality.

Additionally, the mean and variance provide critical information about the distribution’s central tendency and spread. The binomial distribution is a crucial concept in probability theory and statistical analysis.

Its formula allows us to calculate the probability of a certain number of successes or failures in a fixed number of trials and has various real-world applications. We provided examples on how to calculate binomial probability in different scenarios and how to calculate mean and variance.

Understanding the binomial distribution’s parameters and characteristics enables better decision-making in fields such as manufacturing, medicine, and finance. The main takeaway is that the binomial distribution is a useful tool that can help to predict outcomes and make informed decisions based on probabilities.