Law of Truly Large Numbers

Note to my future self…

Today I want to run a short thought experiment after learning about this idea of law of truly large numbers. Before jumping in a couple of quick pointers. First, Law of truly large numbers shouldn’t be confused with the law of large numbers. Second, the law of truly large numbers is simply a formalization of this idea that I wrote about in one of my first posts that volume matters (i.e. the higher the number of at-bats you have the higher the chances of getting lucky). Now that that’s out of the way let’s get a quick definition for the law of truly large numbers:

The law of truly large numbers: is a statistical adage states that with a large enough number of samples, any outrageous thing is likely to be observed. Because we never find it notable when likely events occur, we highlight unlikely events and notice them more.

Example (from Wikipedia)

For a simplified example of the law, assume that a given event happens with a probability for its occurrence of 0.1%, within a single trial. Then, the probability that this so-called unlikely event does not happen (improbability) in a single trial is 99.9% (0.999).

In a sample of 1000 independent trials, however, the probability that the event does not happen in any of them, even once (improbability), is 0.999^1000, or approximately 36.8%. Then, the probability that the event does happen, at least once, in 1000 trials is 1 − 0.9991000 ≈ 0.632 or 63.2%. This means that this “unlikely event” has a probability of 63.2% of happening if 1000 independent trials are conducted, or over 99.9% for 10,000 trials.

The probability that it happens at least once in 10,000 trials is 1 − 0.99910000 ≈ 0.99995 = 99.995%. In other words, a highly unlikely event, given enough trials with some fixed number of draws per trial, is even more likely to occur.


What’s brilliant about this idea is that it deconstructs highly unlikely events into two variables (probability of occurrence and number of trials). Therefore, if you want to experience a highly unlikely event (e.g. building a unicorn company) you should either work really hard to increase the probability of success through things like a great team, capital, GTM strategy, etc. AND/OR increase the number of times that you try to build a great company.

Occurrence of unlikely event = % of occurrence x # of trials.

This model also shows how focusing on increasing the likelihood of success of our company is a lot more valuable than just trying to do 100 different things. That’s evident but I appreciate seeing how it ultimately correlates with our chances of success.

In any case, this idea of truly large numbers is a great way to consolidate what we already. It’s hard to do truly unlikely things but it isn’t impossible. We just have to stay in the game(:

Now back to work…