Binomial Distribution

The discrete probability distribution that an experiment results in only two possible outcomes, that is, success or failure, is known as the binomial distribution in probability theory and statistics.

Let understand it with an example.

1. When a coin is tossed, there are only 2 possible outcomes, which are heads or tails.
2. Similarly, if a test is administered, there are only two possible outcomes: pass or fail.

This distribution is also known as the binomial distribution.
 

The binomial distribution differs from the normal distribution in several key ways. The following table highlights the main differences and traits and key differences between binomial and normal distributions.
 

When the following criteria are met, the binomial distribution can be applied.

• Fixed number of trials: An experiment, such as flipping a coin ten times, has a fixed number of trials, denoted by n.

• Only two possible outcomes: Each trial has two possible outcomes, often called “success” and “failure”. For example, flipping a coin results in either heads or tails.

• Independent trials: Each trial’s verdict is distinct from the others, so the outcome of one does not influence the outcome of another.

• Constant Probability: For every trial, the chance of success (represented by p) stays constant. When you flip a fair coin, for example, the probability of getting heads is always 0.5, meaning there’s an equal chance of getting heads or tails. 

Properties of Binomial Distribution

1. There are two possible outcomes: yes / no, success / failure, or true / false.
    
2. There are  “n” independent trials, meaning the experiment is repeated a fixed number of times under the same conditions.
    
3. Each trial has the same fixed probability of success, and therefore, the same probability of failure.
    

The binomial distribution, formula for a random variable X is given by :
\(P(X=k)=({n \over k}) p^k (1-p)^{n-k}\)

Where,

• n = number of trials
• k = number of successes
• p = probability of success on a single trial
• 1 - p = probability of failure
• \({n \over k} ={ n! \over k!(n-k)!}\) is the number of combinations

For a given probability of success, a binomial distribution table shows the odds of obtaining varying numbers of successes in a predetermined number of trials.

• n is the number of trials.
• p is the likelihood of success.
• P(X = x) is displayed in the table for all x values between 0 and n.

For example, (n = 4, p = 0.5):
 

There is a 0.3750 chance of achieving precisely two successes in four trials.
 

The binomial distribution graph shows the possibilities of obtaining different numbers of successes (X) in a given number of trials (N), which makes it a useful visualization tool.
 

• In the graph below, the distribution plot determines the probability of rolling exactly 0 sixes, 1 sixes, 2 sixes, 3 sixes, ..., and up to 10 sixes in the 10 dice rolls.
   
• This method allows the binomial distribution graph to include the entire range of potential successes up to the total number of trials.

• Each bar in the chart shows the likelihood of rolling a particular number of sixes out of ten dice rolls. Because the probability of rolling seven or more sixes in ten dice rolls is very low, the graph does not show those values.

• According to the binomial distribution graph, there is a roughly 16% chance of rolling no sixes. Rolling exactly one six in ten fair dice rolls is the most likely outcome, with a probability of about (32%).

• However, it often happens to roll two sixes. After three sixes, the odds always decrease. Furthermore, the three sixes bar corresponds to our previous result of 0.155095.
   

In statistics, the binomial distribution is used to model the number of successes in a predetermined number of independent trials, every trial has only two possible outcomes: one is the probability of success or the probability of failure.

Formula:
\(P(X=k)=({n \over k}) p^k (1-p)^{n-k}\)
 

Where,

• X = The number of successes
• n = The number of trials
• p = The probability of success
• n/x = The number of combinations

For example,
The binomial distribution can be used to calculate the probability of getting exactly three heads when flipping a fair coin five times.

The number of trials or observations in a binomial distribution applies when each trial has the same probability of success, and is interested in the number of successes over a fixed number of independent trials. In other words, it calculates the probability of a certain number of successes in a fixed number of trials.
 

A binomial distribution’s mean is represented by 𝜇 = np, where

1. n is the number of observations and
2. p is the likelihood of success.

• The distribution is symmetric about the mean for the instant p = 0.5. The distribution is skewed to the left if p > 0.5 and to the right if p < 0.5.
   
• A binomial distribution’s variance is represented as:
  Variance \(σ^2=npq \ or\ σ^2=np(1-p)\)
   
• A binomial distribution’s standard deviation is represented as:
  Standard Deviation \(σ = \sqrt{ npq} \ or \ σ = \sqrt{ np(1-p)}\)
   
• The following formula provides the coefficient of variation is represented as:
  Coefficient of Variation \(= \sqrt{q \over np} \ or \ \sqrt { {(1-p) \over np} }\)

Binomial Distribution Standard Deviation
The standard deviation is a common measure used to determine how the numbers are from the mean value.

Standard deviation = (Variance)^1/2
                               = (npq)^1/2
 

The central tendency of the dataset can be determined using three key measures: mean, median, and mode.

• Mean
  
  The mean denotes the average value of the dataset. The mean can be determined by dividing the total of all values in the dataset by the quantity of values. It is generally regarded as the arithmetic mean. Additional metrics of mean, utilized to ascertain central tendency, include the following:

1. Geometric mean 
2. Harmonic mean
3. Weighted average

Formula: \(Mean = {x_1 + x_2 + .. + x_n \over n}\)
 

• Median
  
  The median is the central value of a dataset, arranged in either ascending or descending order. When the dataset comprises an even number of values, the median is determined by calculating the mean of the two central values.

Examine the provided dataset comprising an odd number of observations organized in descending order: 24, 22, 19, 17, 16, 14, 13, 11, 10, 8, 7, 6, and 3.

The number 13 is the median, with 6 values above and 6 values below it.
 

• Mode
  
  The mode denotes the most frequently occurring value within the dataset. Occasionally, the dataset may exhibit multiple modes, while in certain instances, it may lack any mode entirely.

Let's practice mode.

Examine the dataset: 6, 5, 3, 4, 3, 2, 6, 5, 6

Let us arrange it in descending order 6, 6, 6, 5, 5, 4, 3, 3, 2

The mode represents the most common value occurring in a data set. Hence, the most frequently repeated value in the given dataset is 6.
 

The number of trials required to reach a specific number of successes in a series of independent trials, where the probability of success in each trial is constant, is known as the negative binomial distribution.

Think about a scenario where the outcome of a die toss is 5. Now, it's a failure if we roll a die and don’t get five. We throw again now, but we don’t get 5. We throw again now, but we don’t get 5.

The negative binomial distribution models the number of trials needed to achieve a fixed number of successes, like getting five on a die multiple times.

For example, if we don’t get 5 for three consecutive tries and five is obtained on the fourth attempt or more, then the binomial distribution of the number of times we get 5 is known as the negative binomial distribution.

Formula
\(P(x)=C_{r-1}^{n+r-1}p^rq^n\)

Where,

• n = Total number of trials
• r = Number of trials in which we get the first success
• p = Probability of success in each trial
• q = Probability of failure in each trial
   

There are two possible outcomes, such as yes / no, head / tail, or even / odd. This means there are two possible outcomes, one that represents the event occurring and one that does not. If a random experiment meets the following criteria, it is referred to as a Bernoulli trial:

• Trials are finite in number 
• Trials are independent of each other
• Each trial has only 2 possible outcomes
• The probability of success and failure in each trial is the same.
   

Two possible outcomes, such as “success” and binomial “failure”, can be used to define a binomial random variable. Take, for example, rolling a fair six-sided die focuses on which number appears, not just the value of the face.

The probability of success for binomial distributions can be determined using the binomial distribution formula. It frequently says to “plug” in the numbers into the formula, to calculate the required values.

The following features serve as the foundation for the binomial distribution:

• There are n identical trials in the experiment.
• One of the two outcomes, the success or failure, occurs after every trial.
• From trial to trial, the probability of success is represented by the letter p, that stays constant.
• Every trial is independent.
   

Students might find binomial distribution difficult and confusion. So here are some tips and tricks to help you understand better.

1. You can use a binomial distribution calculator to verify your answers.
2. Practice solving all the example by yourself.
3. Memorize all binomial distribution formulas.
4. Learn the properties of binomial distribution.
5. Know the relation between mean, median and mode.

Parent Tip: Encourage your child to ask for help when they are stuck. You can do little quizes to help them memorize the formulas.

The binomial distribution is used in real life to predict outcomes and more. Let us see how binomial distribution helps.

1. Manufacturing quality control
   To help guarantee product quality, the binomial distribution is used to calculate the probability of finding a certain number of defective items in a sample.
    
2. Clinical trials
   The binomial distribution is used to estimate how often a new medication is successful by calculating the probability of a certain number of positive outcomes in clinical trials.
    
3. Advertising Campaigns
   Companies forecast the number of consumers who will open a marketing email or reply to an advertisement using the binomial distribution.
    
4. Analytics for sports
   The number of successful attempts a player may make, like a basketball free throw, is estimated by coaches and analysts using the binomial distribution.
    
5. Elections and Polling
   To determine the proportion of survey participants who favor a specific candidate or policy, pollsters use the binomial distribution.