Notes: Binomial Distribution | Applied Mathematics for Class 12 - Commerce PDF Download

Introduction

• Starting with an example, if someone tosses the coin then there is an equal chance of outcome it can be heads or tails. There is a 50% chance of the outcomes. Likewise, if you are appearing in an exam then there is also an equal possibility of getting passed or fail.
• The binomial distribution summarized the number of trials, survey or experiment conducted. It is very useful when each outcome has the equal chance of attaining a particular value. The binomial distribution have some assumptions which show that there is only one outcome and this outcome have an equal chance of occurrence.

Binomial Distribution Criteria

The binomial distribution is a common way to test the distribution and it is frequently used in statistics. There are two most important variables in the binomial formula such as:

• ‘n’ it stands for the number of times the experiment is conducted
• ‘p’ represents the possibility of one specific outcome

It is also being used in social science statistics as elementary units for the models of dual outcome variables. You can take the examples of election polls; whether the party ‘A’ will win or the party ‘B’ will win in the upcoming election. Whether by implementing a certain policy the government will get the expected results within a specific period or not.
There are three different criteria of binomial distributions described below which the binomial distributions need to fulfil.

• The number of the trial or the experiment must be fixed. As you can only figure out the probable chance of occurrence of success in a trail you should have a finite number of trials.
• Every trial is independent. None of your trials should affect the possibility of the next trial.
• The probability always stays the same and equal. The probability of success may be equal for more than one trial.

Whenever you are doing an experiment you need to find out whether this distribution is binomial or not, once you figure that out then you can apply the binomial formula and can count the probability.

Bernoulli Trials

• The binomial distribution is the total or the sum of a number of different independents and identically distributed Bernoulli Trials. In this experiment, the trials are to be random and could have only two outcomes whether it can be success or failure. The flipping of a coin is the best example of Bernoulli trials; each trial can only produce one of the two values- heads or tails. Each time you flip the coin there is a 50% probability of the outcome.
• The outcome never affects or influences the other. The Bernoulli distribution is the part of binomial distribution where the number of experiments is one.

Binomial Distribution Examples

Let’s take some real-life instances where you can use the binomial distribution.

• If the WHO introduced a new cure for a disease then there is an equal chance of success and failure. It can either cure the diseases or not.
• If you are purchasing a lottery then either you are going to win money or you are not. In other words, anywhere the outcome could be a success or a failure that can be proved through binomial distribution.

Binomial Distribution – Formula

First formula

b(x,n,p)= nCx*Px*(1-P)^n-x   for x = 0,1,2,…..n.
where : –
b is the binomial probability.
x is the total number of successes.
p is chances of a success on an individual experiment.
n is the number of trials

• n > 0 ∴ p, q ≥ 0
• ∑b(x,n,p) = b(1) + b(2) + ….. + b(n) = 1
• Value of ‘n’ and ‘p’ must be known for applying the above formula. So, we see that the existence of binomial distribution highly depends on the knowledge of these two parameters. This is why it is also called bi-parametric distribution.

Alternate Formula for Binomial Distribution

The first formula of binomial distribution can calculate the possibility of success for the binomial distribution.  By using this formula you can calculate it. It seems easy but it’s not that easy to calculate unless you are using a calculator. The calculator can reduce a lot of your efforts and time.
But if you want to do it manually then it might take some time but you can solve it through the following simple steps. If a coin is tossed 10 times then what is the chances of getting exactly 6 heads?

• And if you are using the formula –  b(x,n,p)= nCx*Px*(1-P)n-x
• The number of trials is 10 (n)
• The odds of success (tossing heads) is 0.5 (p)
• So, 1-p = 0.5, x= 6
• P(x = 6) = ¹⁰C₆ × 0.5⁶ × 0.5⁴ = 210 × 0.015625 × 0.0625 = 0.205078125

Mean and Variance of a Binomial Distribution

Mean(µ) = np
Variance(σ²) = npq

• The variance of a Binomial Variable is always less than its mean. ∴ npq < np.
• For Maximum Variance: p = q = 0.5 and σ_max = n/4.

Solved Example 

Example: 80% of people those who purchase pet insurance are women. If the owners of 9 pet insurance are randomly selected, then find the probability that exactly 6 out of them are women.

Here are the steps.
Find out the ‘n’ from the problem. Here n = 9
Identify ‘X’. X = the number you are asked to search the probability for is 6.
(Divide the formula then it become easy to get the solution) solve the first part of the formula: – n! / (n-X)! X!
Now add the variables = 9! (9-6)!*6! = 84. And keep it aside for further uses.
Now find out the P and Q. P= the probable chances of success and Q= the possibility of failure. As mentioned in the above question p = 80% or 0.8 so, the probability of failure = 1-0.8 = 0.2 (20%)
Now let’s do the second part of the formula. Px = 0.86= 0.262144
Q^(n-x) = 0.2^(9-6) = 0.2³ = 0.008 (third part of the formula)
Multiply the answer you get from step 3, 5, 6 together
8 × 0.262144 × 0.008 = 0.176