Probability measures the likelihood of an event occurring, such as the probability of a fair coin landing on heads (50 percent or 1/2 chance) or a dice landing on the number 4 (1/6 chance).
To illustrate, consider a school with 10 kids, where 3 play soccer. The probability of randomly picking a soccer player is found by dividing the number of soccer-playing kids by the total children. This is denoted as
$$ P(\text{soccer}) = \frac{\text{no. of kids playing soccer}}{\text{total no. of kids}} = \frac{3}{10} = 0.3 $$
In Venn diagrams, the sample space (denoted by $S$) refers to the set of all possible outcomes and the event (denoted by $E$) is a subset of $S$ to which a probability is assigned. Therefore, the probability of an event occurring is the following formula
$$ P(E) = \frac{|E|}{|S|} $$
where $|E|$ and $|S|$ refers to the cardinality (the number of elements in the set) of the event and sample space respectively
Let’s illustrate using another example. Given that we flip a coin 3 times consecutively, what would be the probability that we land 3 heads? One way to calculate this would be to find out all possible outcomes (sample space) and find out the occurrence of landing 3 heads (event).
We get $P(HHH) = \frac{1}{8} = 0.125$
Now that we understood how to calculate the probability of an event, how do calculate the probability of the event not occurring, also know as the probability of the complementary event. Given an event $A$, its complement is denoted by $E'$. In a random experiment, the probabilities of all possible events (the sample space) must total to 1 - that is, some outcome must occur on every trial. For two events to be complements, they must be collectively exhaustive; together filling the entire sample space. Therefore,
$$ P(E') = 1 - P(E) $$
This is known as the complement rule.
Using the previous example, the probability of not landing 3 heads consecutively is given by
$$ P((HHH)') = 1 - \frac{1}{8} = \frac{7}{8} $$