law of large numbers

What is the Law of Large Numbers? (Brief Explanation)

Whether trigonometry comes in use or not, the statistic is at ultimate use. You will figure out the probability of any situation to make the decision by comparing its possible outcomes and stats. One of the most talked-about theorems is the law of large numbers. 

What is it? Is it helpful in business? Who discovered it? What it focuses on? Are there any sub-sections or types of the law of large numbers? 

Here in this article, you will learn about the law of large numbers both theoretically and practically, along with examples.

What is the Law of Large Numbers?

law of large numbers

Source: probabilisticworld.com

The Law of Large Numbers (LLN) has kind of different definitions with the same context as per the subject. Want to know how? For that, you will have to understand the basic meaning of the law of large numbers.

It is basically a statistic and probability theorem, which explains the result of repeating the same trial a larger number of times. The first person who tried to prove the theorem was mathematician Girolamo Cardano (in 16th Century) but never succeeded.

Then later in the year 1713, the Swiss mathematician Jakob Bernoulli proved the theorem of the law of large numbers in his book Ars Conjectandi. This doesn’t end here it was also refined by one of the good mathematicians known as Pafnuty Chebyshev. He was the founder of Petersburg’s mathematical school.

Enough of history, let’s move on to the exact definition of the Law of Large Numbers. This theorem describes if you repeat the same study for the larger number of times, the average outcome will be closer to the expected value.

Example of Law of Large Numbers

law of large numbers

Source: kastatic.org

Let’s understand the Law of Large Numbers theorem with an example. Here we will take a classic dice experiment. The dice has six sides with respective numbers if you roll the dice thrice the average value may not be near the expected value. What can be the expected value from the viewpoint of the Law of Large Numbers?

For the expected value you will have to add (1+2+3+4+5+6), then divide it by (6), i.e., the m/n formula. The answer is 3.5. Hence, EV = 3.5

Let’s say you rolled the dice thrice, the numbers you got are 3,4, and 5, whose average is 4. Still away from 3.5. So the Law of the large number says, when you repeat the experiment larger number of times, you will get the value near to expected one.

Understood the concept? Let’s move forward to the next concept. This was related to statistics and probability, next have something to do with Finance.

How is the Law of Large Number connected with Finance?

law of large numbers

Source: freepik.com

While the Law of Large number is related to outcomes in Statistics, for finance it is related to the business growth rate. The concept may make you feel low if you are a high-class business person. However, it is a fact according to great mathematicians and hundreds of case-studies.

The concept of LLN in finance tells, that as a business grows, it gets difficult to maintain the previous growth rates. To elaborate it you will see the expansion of the company (on a large scale maybe), but when you consider the growth rates you will see the red line. 

The growth rate of the company will decline every next year. There are chances of LLN to consider market capitalization, net income, and revenue.

Let’s understand it better with the practical example. Take Heed!

Practical Example of Law of Large Numbers in Finance

law of large numbers

Source: law.duke.edu

Let’s say there are two companies namely, SFW Ltd. and KGF Ltd. The market capitalization of the company SFW and KGF is $1 million and $100 million respectively. Let’s say the growth of SFW is 50% for this year, it is attainable as it will grow by $500,000.

But the same is impossible for the company KGF as it will have to attain the growth by $50 million to get to the 50% growth rate point. Thus, as company KGF Ltd. will continue to expand the growth rate of the company will decline over time.

Which are the Two Versions of the Law of Large Number?

law of large numbers

Source: richland.edu

Till here, you read about the concept of Law of Large Numbers, hope you understand the concept moving ahead we have still two versions of the LLN to learn about. Those are Weak and Strong Law of Large Numbers. 

Weak Law of Large Numbers (WLLN)

law of large numbers

Source: ytimg.com

Bernoulli’s theorem is also known as the Weak Law of Large Numbers. So the basic LLN is the weak law of large numbers, where the Sample Mean of repeated experiments of independent and identical random numbers is near to the population mean.

Strong Law of Large Numbers (SLLN)

law of large numbers

Source: slideserve.com

The Strong Law of Large Numbers tells that when the sample size grows infinite times the average mean will converge to the one. In 1909, the French Mathematician  Émile Borel experimented with the new ideas of measure theory and to give a precise mathematical model to formulate the SLLN.

Proving the SLLN is way complex and tough than WLLN. The proofs of the SLLN are there in the graduate books but you can find them here.

Also Read: Calculate The Cash Flow Value By Net Present Value Calculator

Key Takeaways on Law of Large Numbers

law of large numbers

Source: stack.imgur.com

Wrapping up the article by creating a short summary of the Law of Large Numbers, here are key takeaways or you can say, things you learned from the article.

#1: The law of large numbers states, as the sample size of the experiment will grow the average outcome will near to the Expect Value.

#2: The Law of Large Numbers is different for both the statistics and finance. For one it is about the outcome, while for others it is related to the growth rate.

#3: The theorem of Law of Large Numbers complies with the true population and not with the small sample sizes. 

#4: There are two versions of LLN, Weak and Strong. WLLN is easier to prove and SLLN is too complicated.

For more finance-related concepts you can have a look at the FinanceShed, and stay tuned for more updates.