Law of Large Numbers Explained to Kids

The Law of Large Numbers Explained to Kids

Let's go back to our 15 year-old student example. She knows that each time she asks her parents to go to movie theater, there is a 50% chance that they will say yes. Since she loves to go to movie theater, she asks her parents every day. Does that means that after two days she would have seen 1 movie? After 4 days, 2 movies? Maybe yes, but most surely no. 

In fact, the variable "YES" to the question "Can I go to movie Theater?" follows a bernouilli(0.5) distribution. After 4 days, the probability that she would have seen 0 movies is not 0. It is in fact roughly 5%. This is not big, but still there is a chance that she has to wait a couple of more days before seeing a movie. The chart below depicts the probability of no YES after 1, 2, 3, 4 ... 10 days.

However, we know that if she repeats the experiment every day, for an entire year, she will most likely have seen 180 movies during the year. This is the Law of Large Numbers. If you repeat the same experiment (asking for movie theater) a lot of times, the proportion of "YES" (the arithmetic mean) will tend to be exactly equal the the theoretical probability of having a "YES" upfront.

In the chart below, I depict the proportion of "YES" after 1, 2, 3, ... 365 days. We see that after a full year, the proportion of "YES" tends to be roughly 50%. However, if the experiment is repeated only a limited number of times, the proportion of "YES" can diverge significantly from 50%.