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Binomial Theorem: Standard Deviation, Related Terms & Properties

The binomial theorem is one of the most frequently used equations in the field of mathematics and also has a large number of applications in various other fields. Some of the real-world applications of the binomial theorem include:

  • The distribution of IP Addresses to the computers.
  • Prediction of various factors related to the economy of the nation.
  • Weather forecasting.
  • Architecture.

Binomial theorem, also sometimes known as the binomial expansion, is used in statistics, algebra, probability, and various other mathematics and physics fields. The binomial theorem is denoted by the formula below: 

where, n N and x,y R 

Source

What is a Binomial Experiment?

The binomial theorem formula is generally used for calculating the probability of the outcome of a binomial experiment. A binomial experiment is an event that can have only two outcomes. For example, predicting rain on a particular day; the result can only be one of the two cases – either it will rain on that day, or it will not rain that day.

Since there are only two fixed outcomes to a situation, it’s referred to as a binomial experiment. You can find lots of examples of binomial experiments in your daily life. Tossing a coin, winning a race, etc. are binomial experiments. 

Read: Binomial Distribution in Python with Real-World Examples

What is a Binomial Distribution?

The binomial distribution can be termed to measure probability for something to happen or not happen in a binomial experiment. It is generally represented as:

p: The probability that a particular outcome will happen

n: The number of times we perform the experiment

Here are some examples to help you understand, 

  • If we roll the dice 10 times, then n = 10 and p for 1,2,3,4,5 and 6 will be ⅙. 
  • If we toss a coin for 15 times, then n = 15 and p for heads and tails will be ½. 

There are a lot of terms related to the binomial distribution, which can help you find valuable insights about any problem. Let us look at the two main terms, standard deviation and mean of the binomial distribution. 

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Standard deviation of a binomial distribution

The standard deviation of a binomial distribution is determined by the formula below: 

 = npq

Where,

n = Number of trials

p = The probability of successful trial

q = 1-p = The probability of a failed trial

Mean of a binomial distribution

The mean of a binomial distribution is determined by, 

 = n*p

Where,

n = Number of trials

p = The probability of successful trial

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Introduction to the binomial theorem

The binomial theorem can be seen as a method to expand a finite power expression. There are a few things you need to keep in mind about a binomial expansion: 

  • For an equation (x+y)n the number of terms in this expansion is n+1.
  • In the binomial expansion, the sum of exponents of both terms is n.
  • C0n, C1n, C2n, …. is called the binomial coefficients.
  • The binomial coefficients which are at an equal distance from beginning and end are always equal.

Source

Coefficients of all the terms can be found by looking at Pascal’s Triangle. 

Source 

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Terms related to binomial theorem

Let us now look at the most frequently used terms with the binomial theorem

General Term

The general term in the binomial theorem can be referred to as a generic equation for any given term, which will correspond to that specific term if we insert the necessary values in that equation. It is usually represented as Tr+1.

Tr+1=Crn . xn-r . yr

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Middle Term

The middle term of the binomial theorem can be referred to as the middle term’s value in the expansion of the binomial theorem. 

If the number of terms in the expansion is even, the (n/2 + 1)th term is the middle term, and if the number of terms in the binomial expansion is odd, then [(n+1)/2]th and [(n+3)/2)th are the middle terms. 

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Independent Term

The term which is independent of the variables in the expansion of an expression is called the independent term. The independent term in the expansion of axp + (b/xq)]n is

Tr+1 = nCr an-r br, where r = (np/p+q) , which is an integer.

Properties of Binomial Theorem

  1. C0 + C1 + C2 + … + Cn = 2n
  2. C0 + C2 + C4 + … = C1 + C3 + C5 + … = 2n-1
  3. C0 – C1 + C2 – C3 + … +(−1)n . nCn = 0
  4. nC1 + 2.nC2 + 3.nC3 + … + n.nCn = n.2n-1
  5. C1 − 2C2 + 3C3 − 4C4 + … +(−1)n-1 Cn = 0 for n > 1
  6. C02 + C12 + C22 + …Cn2 = [(2n)!/ (n!)2]

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Conclusion 

The binomial theorem is one of the most used formulas used in mathematics. It has one of the most important uses in statistics, which is used to solve problems in data science. 

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In what ways does the binomial theorem come into play in everyday life?

In Statistical and Probability Analysis, the binomial theorem is often utilized. It is quite beneficial, as our economy is heavily reliant on statistical and probability analyses. The Binomial Theorem is used in advanced mathematics and computing to identify roots of equations in higher powers. It's also used to prove a lot of key physics and math equations. Weather Forecast Services, Architecture, and Cost Estimation in Engineering Projects also use the binomial theorem.

What does a Pascal's Triangle imply?

Pascal's triangle is a diagram-based alternative to algebraic approaches for calculating the coefficients that emerge in binomial expansions. This can be a simple method of determining the coefficients for a binomial expansion with a small exponent.

Is there a difference between binomial and Bernoulli?

The binomial distribution is a Bernoulli distribution for a single trial, i.e., when the value of n is 1. A Bernoulli trial or Bernoulli experiment is a single success/failure experiment, whereas a binomial process is a sequence of outcomes. The outcome of a single trial of an event is dealt with by the Bernoulli theorem, but the outcome of several trials of the same event is dealt with by the Binomial theorem. When the outcome of an event is required just once, Bernoulli is employed, but the binomial is used when the outcome is required several times.

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