### Expectation

In the last section you learned that fairness relies heavily upon the concept of probability. When you want something to be fair you expect the outcomes to have the same probability.

It should be a good measure of how fair something is if you measure how far off the results were from what you expected.
You rolled a die 60 times and obtained the outcomes below:

The graph below shows the above data.

There is a line at 10 because that is the expected outcome.

Why is 10 the expected value for each number?

If a die is fair the theoretical probability for any number occuring is 1/6 or approximately .167. So with a lot of trials the proportion of 1s, 2s, 3s, 4s, 5s, and 6s, should get closer to 1/6.

Since 1/6 * 60 is 10,10 is the expected number of 1s, 2s, 3s, etc.

Just as this puppy is expecting a treat, you should expect to get similar results for outcomes that have equal probabilities. If the results are way off of your expectation you should become alarmed. But how much is way off or a lot off.

In the next section you will look at how you can take the difference between what you expected to get and what you got and get a value that will give you a rough estimate about how fair your die is. It will also prime you for the CHI-SQUARE!
The next section is about the D-Statistic!