Mental Model: Law of Large Numbers

Imagine flipping a coin a number of times. The first few times, it may land purely on heads. It may seem as if the universe is magically favouring heads. However, if you flipped the coin a large amount of times, it will eventually even out between heads and tails, and land 50% each. 

This is the 'law of large numbers'. 

Here's the break-down:

(a) For the law of large numbers to occur, it must have independent randomly distributed variables. [1]

(b) ‘...as a sample size grows, its mean will get closer and closer to the average of the population as a whole.’ [2]

(c) Only applicable to large numbers, it can take a long time to approach the mean. [6]

Note: Do not confuse the 'law of large numbers' with the ‘law of averages’ and the ‘gambler's fallacy.’

Some Interesting Examples:

1. The Classic Coin Toss

A coin that is tossed, may initially result in a favour of one side, however, overtime it will balance to a probability of ½. [1]

In coin tossing, the law of large numbers stipulates that the fraction of heads will eventually be close to 1/2.

Hence, if the first 10 tosses produce only 3 heads, it seems that some mystical force must somehow increase the probability of a head, producing a return of the fraction of heads to its ultimate limit of 1/2. Yet the law of large numbers requires no such mystical force.

Indeed, the fraction of heads can take a long time to approach 1/2. For example, to obtain a 95 percent probability that the fraction of heads falls between 0.47 and 0.53, the number of tosses must exceed 1,000. In other words, after 1,000 tosses, an initial shortfall of only 3 heads out of 10 tosses is swamped by results of the remaining 990 tosses. [6]

2. In Casinos

Due to the law of large numbers, a casino may lose money on a single spin on the roulette table, but overtime it will trend towards a predictable percentage. [1]

Casinos are well aware of this law. In part, this is the reason why casinos will always beat the players and end up on top, in the long run. 

If you bang one million dollars at your next visit to Las Vegas at the roulette table in one single shot, you will not be able to “ascertain from this single outcome whether the house has the advantage or if you were particularly out of the gods’ favor. If you slice your gamble into a series of one million bets of one dollar each, the amount you recover will systematically show the casino’s advantage. This is the core of sampling theory, traditionally called the law of large numbers. [8]

3. In Large Businesses

The law of large numbers in the financial context has a different connotation, which is that a large entity which is growing rapidly cannot maintain that growth pace forever.

...assume that company X has a market capitalization of $400 billion and company Y has a market capitalization of $5 billion. In order for company X to grow by 50%, it must increase its market capitalization by $200 billion, while company Y would only have to increase its market capitalization by $2.5 billion.

The law of large numbers suggests that it is much more likely that company Y will be able to expand by 50% than company X. [2] 

4. In Sales Forecasting

The law of large numbers states that a series of big numbers will lend to be more stable, less jumpy, less “nervous“ — and thus easier to forecast than small ones.

Have you ever noticed how the sales forecast for the entire company is usually pretty accurate? And that individual item forecasts are oflen all over the place?

This is the law of large numbers at work, and it represents another good way to reduce forecast error. [7]

5. Gathering Evidence

The preference for more evidence seems well understood as being due to an intuitive appreciation of the law of large numbers.

For example, we think that most people would prefer to hold a 20-minute interview rather than a 5-minute interview with a prospective employee and that if questioned they would justify this preference by saying that 5 minutes is too short a period to get an accurate idea of what the job candidate is like. That is, they believe that there is a greater chance of substantial error with the smaller sample.

Similarly, most people would believe the result of a survey of 100 people more than they would believe that of a survey of 10 people; again, their reason would be based on the law of large numbers. [4]

Sources:

I have looked up this theory in a number of books and I found that Wikipedia & Investopedia had superior explanations.

[1] https://en.wikipedia.org/wiki/Law_of_large_numbers

[2] http://www.investopedia.com/terms/l/lawoflargenumbers.asp

[3] Banking and Finance by Britannica

[4] Rules for Reasoning edited by Richard E. Nisbett

[5] Decision Making: Descriptive, Normative, and Prescriptive Interactions by David E. Bell, Howard Raiffa

[6] The Britannica Guide to Statistics and Probability by Britannica Educational Publishing

[7] Sales Forecasting: A New Approach by Thomas F. Wallace, Robert A. Stahl

[8] Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets by Nassim Nicholas Taleb

Founder: Jacob Bernoulli - Swiss Mathematician [3]

Categorisation: statistics → probability

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