Relative Frequency and the Law of Large Numbers

Now you will learn about relative frequency, which is closely related to the law of large numbers.

You have previously learned that the probability of getting a five when you roll a die is $\frac{1}{6}$. But how do you really know it’s exactly $\frac{1}{6}$?

If you roll a die 12 times, you may not get a five exactly one sixth of the time. After 12 throws that would be

$$12\cdot \frac{1}{6}=\frac{12}{6}=2$$

times. You may get five fives, three fives, or maybe no fives.

To get an answer as to why we say the probability of getting a five when you roll a die is $\frac{1}{6}$, you need to learn two new concepts: relative frequency and simulation.

Relative Frequency

Imagine rolling a die 12 times and writing down how many fives you get. The relative frequency is the ratio of how many fives you got (number of favorable outcomes), to how many times you rolled the die (number of trials). If you got three fives, the relative frequency is

$$\frac{3}{12}=\frac{1}{4}=0.25$$

This is a lot more than you expected, which was $\frac{1}{6}$. So let’s look at why.

The relative frequency of getting a five when rolling a die does not need to be the same as the probability ($\frac{1}{6}\ne \frac{1}{4}$). But if you roll a die a lot of times, for example $100000$ or $1000000$ times, the relative frequency of getting a five will be just about the same as the probability of getting a five.

Note! The relative frequency approaches the probability when the number of trials becomes very large!

Law of Large Numbers

The average age of people in the US is about 38 years old. You decide to personally ask a bunch of people how old they are. After you ask each person, you calculate the average age of the people you have asked so far. How many people do you have to ask before you get 38 years old as the average? Maybe it’s sometimes sufficient to ask five people. But what if the five you asked are all a group of friends in junior high? Then you will certainly not find the average age of the population to be 38 years old.

It makes sense that the answer is 38 if you just ask enough people. But how many are enough?

This is what the law of large numbers has something to say about.

Unless you are investigating something very unlikely, then somewhere between $1000$ and $10000$ trials will be enough. It would be difficult to ask $10000$ random people in the US about their age and not get an average very close to 38.

Below, I made a graph of the relative frequency of getting a five when rolling a die. The $x$-axis shows the number of throws, which is the number of trials, and the $y$-axis displays the relative frequency for getting a five. The black line through the middle illustrates the theoretical probability of getting a five when rolling one die, which should always be $\frac{1}{6}.$

You can see from the figure that after approximately $1500$ throws, the blue relative frequency has stabilized very close to the actual probability in black.

Next, let’s look at some examples where it’s important to know the law of large numbers.

Simulation

Rolling a die $1000000$ times to see how many fives you get would be quite tiring and very time consuming. But if you did, you would end up getting about $166667$ fives. That would be a relative frequency of $\frac{166667}{1000000}\approx \frac{1}{6}$, and you can see that this is exactly the same as the probability of getting a five!

To avoid having to roll a die $1000000$ times, you can do a simulation instead, using a computer program to do the experiment for you.

Simulating means that we are imitating an experiment from reality on a computer. We can mimic dice rolls on the computer, rolling a virtual die many, many times over a short period of time. In this way, you’re then able to see what the probability of an event is, with the computer’s help. The computer calculates the relative frequency for such a large $N$ that the relative frequency becomes the same as the probability.

Computers are a very important tool when working with probability.