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The Law of Large Numbers: A Statistical Advantage to Trading Options

The foundation of statistically-based options trades looks to one statistical law – The Law of Large Numbers.

The Law of Large Numbers states that as you increase your sample size (e.g. number of trades), our expected value or probability of success will come to fruition.

This is because the Central Limit Theorem shows us that actual values will converge on expected values.

Trade often applies the Law of Large Numbers/Central Limit Theorem. Because of the Central Limit Theorem, we know how much variability we can expect between our average P&L and the long-term average based on the number of trades we made. The more trades we make, the more likely our average P&L is to be closer to the long-term historical average of all trades.

KEY: More trades = less uncertainty.

If held to 21 days to expiration, the key takeaway is that the volatility of total Profit & Loss goes down significantly meaning you take less risk and are more profitable.

Avoid the gambler’s fallacy that says the past will continue in the future. If the last 9 out of 10 trades are winners, the 10th trade is not guaranteed to be a winner. Trade based on mechanics discussed in this paper. The key point is to trade smaller rather than bigger as you can get overly confident. This is one of the hardest things to master as a trader.

For the Central Limit Theorem to work, we need a large enough sample size or a number of observations – in our case, trades. This is where the Law of Large Numbers comes in.


A coin has a 50% chance, or probability, of landing on heads or tails.

According to the Law of Large Numbers, the number of heads in a large number of coin flips should be 50%, which is known as our expected value.

What about VARIANCE

The Law works over the long term. However, you might see what is referred to as sequencing risk when only flipping the coin 10 times (or some smaller number of flips). When flipping a coin 10 times, the variance of the coin landing on heads has an average range of three heads to seven heads based on standard deviation. As we increase the number of observations the range will tighten until the probability of success approaches 50%.

Over the long term, the expected value is 50%. However, with small sample size, you most likely will not see an even 50/50 split. We call these statistical outliers and they are to be expected.

Make the Law of Large Numbers work for you – trade small, trade often, and be patient.


The Law of Large Numbers is important to understand because, unlike a coin flip that has a 50% probability of success, my approach has a probability of success that is significantly higher, roughly 75% to 90%.

Much of a trader’s success relies on how disciplined you are at mitigating risk via being rules-based, using best practices for your trade size as well as the best times to enter and exit trades. We cover all of these extensively with our subscribers.

If my approach interests you and you have any questions please feel free to email me at

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