What is Binomial Distribution?

Introduction

A binomial distribution can be considered the likelihood of Success or Failure outcome in numerous experiments or surveys: binomial distribution is a form of a probability distribution characterised by two possible outcomes. A coin flip, for example, has only two outcomes: heads or tails, and passing a test has only two outcomes: pass or fail.

In a binomial distribution, the parameters n and p are employed. The variable ‘n’ indicates how many times the experiment will run. In contrast, the variable ‘p’ indicates the probability of each outcome. If a dice is thrown ten times at random, the probability of getting two on any toss is 16%. You get a binomial distribution with n = 10 and p = 16 if you throw the dice ten times. 

What is Binomial Distribution?

In binomial distribution, the number of ‘Success’ in a series of n experiments, where each time a yes-no question is given, the boolean-valued answer is indicated with either a positive outcome (yes/true/one) or a negative outcome (failure/no/false/zero) (probability q = 1 p) in a binomial probability distribution.

The focus of the Bernoulli trial or Bernoulli experiment is on a single success/failure test, and a Bernoulli process is a series of outcomes. The binomial distribution is a Bernoulli distribution for n = 1, i.e., a single experiment. The renowned binomial test of statistical significance is based on the binomial distribution.

In real life, binomial distributions can be abundantly found. When a new drug is presented to cure a sickness, it either cures the disease (success) or does not cure the disease (failure). If you buy a lottery ticket, you’ll either win or lose money. The use of binomial distribution can be done to represent almost anything that can only be a success or a failure.

The Bernoulli Distribution

A set of Bernoulli trials is referred to as a Bernoulli distribution. Each Bernoulli trial can result in two outcomes: success or failure. The chance of success in each trial, P(S) = p, is the same. P(F) = 1 – p = 1 – p = 1 – p = 1 – p = 1 – p = 1 – p = 1 – p = 1 – p = 1 – (Keep in mind that “1” represents the total likelihood of an event occurring…probability is always between zero and one.)

Finally, all Bernoulli trials are independent of one another, and the likelihood of success does not alter from one trial to the next, even if you are aware of the results of previous trials.

Negative Binomial Distribution

Negative Binomial Distribution is simply the number of successes of independent and identically distributed Bernoulli trials in a row before a specific number of failures occur. Negative Binomial Distribution is a concept in probability theory and statistics. The number of failures is indicated by the letter ‘r’. 

For example, suppose we roll a die and declare 1 to be a failure and all other numbers to be successes. If we toss the dice often until 1 appears for the third time, i.e., r = three failures, the probability distribution of the number of non-1s that comes is the negative binomial distribution.

The Binomial Distribution Formula

The binomial distribution formula is

P(x:n,p) = nCx x px(1-p)n-x 

OR

P(x:n,p) = nCx x px(q)n-x

Where,

n is the number of experiments

the value of ‘x’ is 0, 1, 2, 3, 4, …

p is the probability of a positive outcome (success) in a single experiment

q is the probability of a negative outcome (failure) in a single experiment = 1 – p

Binomial Distribution Mean and Variance

Binomial distribution’s mean, variance and standard deviation for the given number of successes can be expressed with the help of the following formulas.

Mean, μ = np

Variance, σ2 = npq

Standard Deviation σ= √(npq)

Where, 

p – the probability of success

q – the probability of failure (q = 1-p)

Binomial Distribution Examples

1- A coin is tossed 5 times. What is the probability of getting heads 3 times?

• Let x be the number of heads. In this case, x is 3. 

The number of trials (n) is 5

The probability of success (“tossing a heads”) is 0.5. So, 1-p = 0.5

Binomial distribution formula used: P(x:n,p) = nCx px(1-p)n-x 

Therefore, P(x=3) = nCx 0.53(1-0.5)5-3 

x= 5/16

2- Men account for 60% of those who buy sports cars. Find the likelihood that 7 of 10 sports vehicle owners are guys if they are chosen at random.

Step 1: From the problem, find ‘n’ and ‘X.’ In our example, n (the number of randomly picked items—in this case, sports car owners) is 10, and X (the number for which you are requested to “determine the likelihood”) is 7.

Step 2: Determine the first part of the equation, that is:

n! / (n – X)! X!

Changing the variables to: 10! / ((10 – 7)! 7!)

This adds up to 120. Now, keep this number aside

Step 3: Calculate “p” for the chance of success and “q” for the chance of failure. Because p = 60%, or.6, the likelihood of failure is 1 –.6 =.4. (40 percent).

Step 4: When we work on the next step in the above-mentioned formula.

pX

= .67

= .0.0279936

Set this figure apart while you work on the formula’s third part.

Step 5: When we work on the last part of the formula.

q(.4 – 7)

= .4(10-7)

= .43

= .0.064

Step 6: Multiply the three answers from steps 2, 4, and 5 together.

120 × 0.0279936 × 0.064 = 0.215.

Conclusion

Getting a grasp of the concept of binomial distribution can go a long way. When a process is repeated a certain number of times (e.g., in a group of patients) and the outcome for each patient is either a success or a failure, the binomial distribution model allows us to calculate the chance of seeing a certain number of “successes.”