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The Law of Large Numbers (Another Mental Model)

The law of large numbers indicates that the higher the number of times something is performed, the more likely it will receive something close to the average of the results.

For example, if you rolled a six-sided die, the average results you can expect is the average of the 6 outcomes is a 3.5 ( (1 + 2 + 3 + 4 + 5 + 6)/6. The law of large numbers indicates that your average roll will get closer and closer to 3.5 the more you roll the dice!

That’s why when you’re winning at a casino (or anywhere else where “the house” takes its cut), you should quit early — otherwise, the law of large numbers will eventually kick in: the more bets you make the closer you will get to netting a loss (since a casino is designed to have a greater than 50/50 edge over you).

But don’t fall victim to what’s called the “gambler’s fallacy”. In the gambler’s fallacy, if you flip a coin 5 times, and it comes up heads, some folks falsely believe that then the 6th toss is more likely to come up tails. That’s not really true. Each single event is still a 50/50 probability. The law of large numbers instead says that the more you flip the coin the closer the average result will be to heads/tails being at a 50/50 split.

For example, if you tossed a coin just 10 times, you could easily get 8 heads (80%) and 2 tails (20%); but if you tossed a coin 1,000 times, the law of large numbers smooths out the results so that the heads/tails results would be closer to the average of 50%.

As you take more and more samples, the average of that sample will converge to the expected value.

The law of numbers is also used to explain how the larger the number you have the more you have to work to grow that number.

For example, if Facebook has $1 billion in profits and Berkshire Hathaway has $10 billion in profits, the law of numbers indicates that Facebook will have an easier time growing profits 10% per year over the next 1o years  than Berkshire could (all other things equal) because Berkshire’s starting base is so much larger.

This article on Attribution Theory continues my quest to list out The Top 100 Mental Models Needed to Succeed in Business.