# Empirical Probability

## Empirical Probability

Hello and welcome to this video about **empirical probability**! In this video, we will explore what empirical probability is, how empirical probability relates to **theoretical probability**, and how empirical probability can be used.

Let’s use a standard number cube to introduce some of the key concepts related to empirical probability. A standard number cube has 6 sides, numbered 1-6. The weight is equally distributed, and the sides are the same size, so the theoretical probability of rolling any number is 1 out of 6, as illustrated here:

The theoretical probability of rolling any number is 1out of 6 because there are 6 equally possible, **independent outcomes** every time the number cube is rolled. In theory, after, say, 96 rolls, each number should be rolled 16 times.

So you grab a number cube and start rolling, but your outcomes look like this:

Clearly, these outcomes didn’t match their theoretical probabilities. Why? Theoretical probability simply states the **likelihood of an outcome** of an event, but it does not state what must happen. The **frequencies** of the outcomes you rolled are your empirical probabilities (also called **experimental probabilities**). If you rolled the number cube a 97th time, you might predict an outcome of a 4 or a 5 because the combined empirical probabilities of these two outcomes are 27 out of 96 plus 31 out of 96, equals 58 out of 96.

This illustrates the key distinction between theoretical probability and empirical probability. Theoretical probability of an event can be calculated without actually doing the event. There are 6 possible outcomes of rolling a number cube, so each outcome has a theoretical probability of 1 out of 6. There are 4 aces in a deck of cards, so the probability of randomly drawing an ace is 4 out of 52. On a standard dartboard, the area of the double bullseye in the center is 0.078% of the area of the dartboard, so the probability of a double bullseye is about 78 out 100,000. On the other hand, empirical probability uses the frequencies of outcomes to determine the likelihoods of future outcomes. They are just **estimates**. When you rolled the number cube, 4s and 5s are what you rolled most, so the empirical probability of rolling a 4 or a 5 on the next roll is 58 out of 96.

It’s also important to note that theoretical probabilities **don’t change**. The theoretical probability of rolling any number on the number cube is always 1 out of 6. Suppose you decide to try the experiment again the next day and get these results:

The empirical probabilities here are very different and you might come to a different conclusion than in the first round.

Combining the results of the two experiments might lead to a third way of thinking:

This emphasizes the use of empirical probability as simply a gauge. Knowing some past outcomes can help guide in predicting future outcomes. Again, empirical probabilities are just estimates. In this case, the differences in the outcomes reinforce the fact that rolling a number cube is an independent event and that any outcome has an equal probability of occurring.

Of course, according to the **Law of Large Numbers**, if you rolled the number cube 1,000 or 10,000 times, the empirical probabilities would start to more closely resemble the theoretical probabilities.

So we’ve talked about the numbers you rolled the most. What about the number you rolled the least? Over 192 rolls, 6 was rolled the least. That doesn’t mean that you are “due” for a 6 soon. The probability of rolling a 6 is still 1 out of 6 and the event is still independent. In other words, the theoretical probability does not become higher on the 193rd roll because the empirical probability from the previous 192 rolls is low.

When meteorologists predict the weather, they are largely using empirical probabilities. When a weatherman says there is a 70% chance of rain tomorrow, that number isn’t arbitrary. Part of what it means is that certain weather conditions are expected tomorrow that could lead to rain. Historically, when similar conditions have been present, it has rained about 70% of the time. The likelihood of the outcome of a future event is based on the outcomes of similar past events.

I hope that this video increased your understanding of empirical probability and how it can be used!

See you next time!