The Law Of Large Numbers In Trading
In order for us to be a successful trader in the market over the long run, we have to understand the concept of the law of large numbers and how it benefits us to apply this concept to the way we look at our trades
The law of large numbers demonstrates and proves the fundamental relationship between the concepts of probability and number of frequency. In a way, it provides the bridge between probability theory and the real world
Definition of the Law of large numbers
It’s a theorem that describes the result of performing the same random experiment a large number of times. According to the law, 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. [Wikipedia]
The same random experiment needs to be repeated a large number of times means that the experiment is repeated under identical conditions and the probability of each outcome is independent of any of the previous outcomes
Let’s look at some examples of the application of the law of large numbers
Law of large numbers with Coins: Imagine we have these special coins where both sides of the coin are the same (either both are heads or both are tails). Say we have exactly 50 of the “heads” coins and 50 of the “tails” coins and we put all 100 coins in a bag and mix it really well
Now we close our eyes and draw N number of those coins and calculate the % of them that were heads. This should be familiar territory by now
If drawn N = 1, the % will be 0 or 100, depending on the kind of coin we happened to draw
What if drawn N = 99?
Now we have drawn all but 1 of the coins. If the one remaining coin is a “heads” coin, the proportion of heads among the drawn coins will be 49/99, which is about 49.5%
On the other hand, if the remaining coin is “tails”, the proportion will be 50/99, or approximately 50.5%
“In both cases, the % is very close to the actual 50% of “heads” coins in the bag”
What if we make N = 98?
Here, the worst possible deviation from 50% would happen if the 2 remaining coins were of the same kind
In those cases, the % of heads would be either 48/98 = 49% or 50/98 = 51%
The point with this is that, when number drawn is very close to 100, even in the worst case scenario we still get a relative frequency very close to the real one (50%)
By following this procedure, we are essentially creating the identical and independent conditions required by the law of large numbers
Law of large numbers application in Casinos: In regards to how the casinos operates with the law of large numbers, they might have one or two players that win a large amount but over a long period of playing time and a huge amount of players, the house always wins
It doesn’t matter if one or two players win(casino losses) but overtime through a large number of data the house will always wins
Law of large numbers in insurance company: Insurance companies use the law of large numbers to lessen their own risk of loss by pooling a “large enough number” of people together in an insured group. The size of the pool corresponds to the predictability of the losses
For example, an auto insurance company may record and study the number of accidents caused by a very large population of 18-year-old males. So that they will be able to predict the likelihood of how many 18-year-old males will cause an accident in a given year
With the statistics of a large population, they will know that in a given year there is a high probability that X number of 18-year-old males is likely to cause an accident. Knowing this, they partially can determine how much an 18-year-old male should pay for auto insurance (excluding other factors, such as the type of vehicle, region where the driver resides, etc.)
This is how the law of large numbers helps insurance providers determine their rates, and why the rates vary from one type of individual to another
The larger the population calculated from statistics, the more accurate the predictions
Law of large numbers applications in trading: No matter the expectancy of our strategy, trades are randomly distributed. They don’t come in 1 win, 1 loss, 1 win, 1 loss or however we’re expecting the distribution. For that reason, we can’t predict in advance if the next trade is a losing trade or a winning trade
We just have to accept that in the outcome of one single trade, there’s complete randomness. But the good news is that over the course of a large number of trades, there’s statistical order with any chosen strategy
Remember that the overall outcome of a trading journey does not unfold linearly. It’s not a straight line going up
We literally have clusters of winning trades and clusters of losing trades, and we don’t know when these clusters will appear. So we’re not supposed to look at our trading outcomes in a “certain small period” and extrapolate them into the future; which simply means we should not be discouraged by losing streak or carried away by winning streaks
With a fairly good strategy in the long run, winning and losing streaks tend to "even out" each other and it’s the long run distribution of winnings and losses that makes us profitable because the long run performance have very little to do with luck factor. In other words, trading the short term basically have to do with luck which is the gambling mindset but the long game is all about “statistics and probability” that will guarantee our success
The law of large numbers guarantees stable long-term results for the averages of some random events
How do we describe large enough numbers?
The short answer is that the question itself is a bit vague. A piece of information that needs to be added to the above question is how close we need to get to the expected frequency
For example, when flipping a coin, if you might not mind a difference of, say 5%; In that case, even a few flips will get you there. On the other hand, if you need to rely on a relative frequency that is almost equal to the expected 50%, you will need a high number of flips. Like the law says, “the higher the number of trials, the closer the relative frequency will be to the expected one”
The law of large numbers shows the inherent relationship between relative frequency and probability. In a way, it is what makes probabilities useful. It essentially allows people to make predictions about real-world events based on them
The law of large numbers is a mathematical theorem but it’s probably not an incident that it actually has the word ‘law’ in it. Let’s think about it in similar terms as some natural physical laws, such as gravitation. The probability of the outcome is like a large body that pulls the empirical frequency towards itself
The exact trajectory might be different every time, but sooner or later it will reach it. Just like a paper plane thrown from the top of a building will eventually reach the ground
Trading is a BUSINESS - Treat it as such!!
