Small numbers, big questions

One of the first didactic elements for the study of probability and statistics involves the use of coins.

When flipping a coin, the outcome can be “heads” or “tails”, each with a probability of 50 percent. With the intention of not generating ambiguities, sometimes emphasis is placed on the “fairness” of the coin.

However, some skeptics may take upon the task of empirically testing the aforementioned odds, and end up doubting the “fairness” of their coin after getting 8 “heads” out from 10 flips, or other highly concentrated outcome that make the 50/50 rule sound dubious.

But, how can this happen?

The law of large numbers describes that the result of repeating an experiment more times will approximate the results to the expected value. In other words, the more times you flip the coin, the distribution of results will get closer and closer to 50/50.

Now let’s go to roulette, a person who bets on a specific number can earn 35 dollars for every dollar wagered. This would seem fair to anyone who considers that the only options range from 1 to 36; however, American Roulette includes options 0 and 00, giving a 5.26% edge to the house.

Hence, the way in which a casino guarantees that this advantage materializes is by increasing the iterations, aiming for more people playing more times. This makes negligible the fact that, eventually, someone can earn a hefty profit. On the long run, the advantage is on their side.

If you have an edge, how can you guarantee to profit from it?

By studying the returns of the market in the long term, and assuming that the distribution of returns shows normality, we can define as expected value a return of 7.6 percent per year in dollars, like the one posted by the S&P 500 in the last 100 years.

In the same way that we would get a 50/50 distribution by flipping more coins, and that a casino can beat roulette players, the return in the stock market will be your friend the longer you manage to stay in it.

Following the Law of Large Numbers, we can also see that the variance—the dimension with which the results diverge from the average—is usually greater if the sample is small. For this reason, making inferences with few observations will deliver expectations biased by the presence of large-scale variability.

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