The average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed (the law of large numbers is true).

Understanding the Rule of Large Numbers

The law of large numbers says, the more instances of a probable event that are considered, the more the theoretical and actual results converge.

For example, the more times you flip a coin, the more likely it will be that heads and tails both come up exactly 50% of the time on average (specifically, that the mean result will approach 50%). Over an infinite amount of instances (in the case of a coin, over an infinite amount of coin flips), the real and expected results coverage, approaching the theoretical odds (they approach convergence, they don’t necessarily converge perfectly).[1]

This doesn’t mean the probability of single coin flip changes (see the gambler’s fallacy), it means that the mean results approaches convergence with the theoretical results as the instances of coin flips increase.

This is why a single spin at roulette is pure chance, but the casino can fairly accurately calculate their return from all spins that will ever be spun over the life of the wheel.

This law of large numbers applies to many things including:

  • The average payout of slot machines can be calculated, and then programed into the machine, to ensure a return for the house, over countless machines, over countless spins.
  • Guessing the weight of a person, or the number of jelly beans in a jar, at a carnival game (it applies in unbiased democratic situations, the more guesses, the more likely the mean answer is right… but only when guesses are fact-based, thus propaganda and special interests make pure direct democracy tricky, politically speaking).
  • Calculations for insurance adjustments, risk can be calculated for all members, of all plans, to set a rate (and rules) that are favorable to the insurer.
  • Determining the reflective properties of a substance.
  • Determining the quantum behavior of elementary particles. We can calculate where in the probability wave a photon will appear.
  • Determining where a shale deposit will be with greater accuracy to limit some of the guesswork involved in mining (i.e. a common use is for big-data and big-business).
  • And more. We can apply the rule to anything that involves “chance” to know the odds we calculate will come closer and closer to manifestation as we increase trails, even if we can’t (or don’t want to) do a test run.

The world is ruled by probability (on a quantum, and consequently physical and mathematical, level). Since the rule of large numbers allows us to predict complex random behavior using mathematics (rather than relying on physical trials and empirical evidence, and rather than randomness being truly indeterminable), it is generally a vital rule in probability theory.

The concept is a little tricky to grasp from just an explanation, so consider watching the following video for a visual.

Law of Large Numbers – Explained and Visualized. This video explains the law of large numbers as it applies to a range of fields.

TIP: The rule of large numbers should not be confused with the gambler’s fallacy. Each independent event is unique, the rule only applies when considering many instances. A fair coin always has a 50/50 chance of being heads or tails, regardless of past results, so this rule won’t help you at the roulette table in the short term… but it will allow you to calculate near perfect odds over an infinite amount of spins.

TIP: If a photon will reflect off of a theoretical surface 25% of the time, it doesn’t mean 1 in 4 photons bounce away. It means that over an infinite amount of instances, 25% of photons will reflect. This is the rule of large numbers in play on the “realest” of levels, the quantum level.

FACT: The law of large numbers was first proved by the Swiss mathematician Jakob Bernoulli (the Bernoulli numbers guy) in 1713. He and his contemporaries were developing a formal probability theory based on analyzing games of chance (AKA gambling games). Funnily enough, Pascal, another famous mathematician, invented probability theory based on gambling as well. Oddly, or not, this vital mathematical system is very useful for figuring out betting odds. It works so well that casinos can rely on random number generators and algorithms based on large numbers for their profits. The law of large numbers is essentially what keeps a casino’s lights on and explains how those lights reflect off the Roulette wheel as well.

Example of the Rule of Large Numbers and Dice

This example from Wikipedia provides most of the information that you need to know to grasp the concept behind the law of large numbers:

{\frac {1+2+3+4+5+6}{6}}=3.5.

The expected value of a single die roll is 3.5. According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the sample mean) is likely to be close to 3.5, with the precision increasing as more dice are rolled.[2]

In the same way, with a coin flip, at any point, the actual number of heads may be more or less than tails, but the proportion of heads after “x” amount of flips will almost surely move toward 1/2 as “x” approaches infinity!

TIP: Volatility describes the distribution of results. We can look at volatility and the standard deviation from the mean to get a sense of how likely results will converge in a given time frame.

FACTDiffusion is an example of the law of large numbers, applied to chemistry.


The rule of large numbers, its not just a good idea, it is the law.


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"The More Instances Considered, The More Theoretical and Actual Results Converge" is tagged with: Blaise Pascal, Gambling, Probability and Odds, Quantum Mechanics, Randomness, Theoretical Physics, Theories

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