The Law of Large Numbers

The law of large numbers states that the more experiments that occur, the higher the chances are that the outcomes will trend toward whatever the probability is.

For example, let’s say that you are in a coin flipping contest and there are three rounds. The contest has two players and it is one-on-one. The probability of either contestant getting a heads or a tails is 50%.

Now let’s say that in each round the side of the coin that won was heads. So there were three heads in a row. Over three rounds, heads winning each time is pretty likely because the amount of rounds (or experiments) isn’t very large. In other words the sample size is small.

But what if there were twenty rounds instead of three? The chances of heads occurring twenty times in a row is possible, but very, very unlikely. So over twenty rounds there will be more tails occurring and this will cause the amount of tails to trend towards occurring ten times which is what the probability of flipping a coin is - 50%.

So the law of large numbers just simply states that the more an event occurs, the more the chances of that event occurring will trend towards its average over time.

The law of large numbers applies in a lot of different areas in the real world. Two areas are sports and insurance

When a .300 hitter in baseball goes on a hot streak and gets six hits in ten at bats during a week, you can likely assume that he will only get three hits out of ten and bat .300 the following week or maybe even get 0 hits which will bring him right back to his average of .300.

The insurance industry lives on the law of large numbers. They are all about managing risk and pricing it as accurately as possible. A small sample could bankrupt an insurer but if they insure lots of policies where they can get a good enough estimate of the probability of a certain type of risk occurring like a hurricane or a car crash, and then price that insurance policy adequately for the risk, and then be patient and disciplined, then they can make good profits over a number of years.

I flipped a coin 10 times to demonstrate the law of large numbers. Below are the results. As you can see after the first 4 coin flips there were 3 Heads and 1 Tails which is an outcome of 75% Heads and 25% Tails. I then flipped the coin 6 more times for a total of 10 flips and there were 5 Heads and 5 Tails which is an outcome of 50% Heads and 50% Tails - the exact probability of each outcome.

Flip #1: Heads

Flip #2: Tails

Flip #3: Heads

Flip #4: Heads

Flip #5: Tails

Flip #6: Heads

Flip #7: Tails

Flip #8: Tails

Flip #9: Heads

Flip #10: Tails