We’ve already seen this formula showing us how to calculate the probability of an event occurring:

                               

The top part of the fraction is often referred to as the number of favourable outcomes – the number of outcomes resulting in the event we’re interested in. 

The number of favourable outcomes can be thought of as a set of outcomes.  We can use the mathematical language of sets to rewrite the top of the equation as:

                                

In this simple mathematical expression, ‘E’ is the event that we are calculating the probability for.  The ‘n(E)’ part can be translated as, “the number of things in set E.”  In other words, n(E) is the number of outcomes resulting in the event we’re calculating the probability for.

The bottom part of the fraction can also be rewritten.  The bottom part is concerned with all possible outcomes.  This can also be thought of as a set – a set containing all the possible outcomes in the experiment.  We can call this set A, with ‘A’ standing for “All”.  The bottom of the fraction becomes:

                                      

This time the ‘A’ stands for the set of (A)ll possible outcomes.  We can rewrite the entire formula like this:

                                                      

Subsets

The set E contains the outcomes that result in a favourable event.  The set A contains all the possible outcomes, including the favourable ones.  This means that the set E is a subset (smaller part of) the set A.  The formal mathematical way of writing this is like this:

                                                            

The  symbol means “is a subset of”. 

‘Not’ sets

Say we have a set E representing all the outcomes that result in the event E.  This set can simply be represented by ‘E’.  What about if we want to talk about the set of outcomes that don’t result in the event E?  Before, we talked about this in terms of “E” and “Not E”.  Well, using proper mathematical notation, we can write “Not E” as:

                                                               

Complementary sets

The probability of event E occurring and the probability of it not occurring adds up to 1, or 100%.  We can talk about these as two sets. Set E contains all the outcomes that result in E happening.  The other set, , contains all the outcomes that don’t result in E happening. 

Now because the probability of these two sets happening adds up to 1, we can say that these are ‘complementary’ sets.  They complement each other to make up a total probability of 1.