The probability of an event \(A\) is the proportion of the sample space \(S\) where the event \(A\) occurs. Mathematically, this is:
$$P(A) = \frac{\text{size of $A$}}{\text{size of $S$}}$$
The complement of an event is the proportion of the sample space where an event \(A\) does not occur.
$$P(\bar{A}) = 1 - P(A)$$
Note that the probability of the event and its complement occurring is zero, since the sets are disjoint (have no intersection):
$$P(A \mid \bar{A}) = 0$$
When two events occur together, the probability of both events occurring is the size of the intersection of both sets. This is written as:
$$A \cap B$$
This is usually given to you in a table. However, it can also be calculated by:
$$P(A \cap B) = P(A) P(B \mid A)$$
This works, because the only place where the intersection of \(A\) and \(B\) would lie in some portion of \(A\).
You can combine the probability of two events happening, \(A \cup B\), by using the following rule:
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
You can find the proportion of values that are within a subset \(C\) of \(A\):
$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$
This would be used to answer a question: "what proportion of guys are late for class?" Notice how this is different from the question "what proportion of people are both guys and late for class?"
Two events \(A\) and \(B\) are mutually exclusive or disjoint if their intersection is empty. This means that they do not contain any of the same outcomes, and \(P(A \cap B) = 0\).
Two events are independent if knowing that one occurs does not change the probability that the other occurs.
$$P(A \mid B) = P(A)$$
$$P(B \mid A) = P(B)$$
There is also a special probability multiplication rule:
$$P(A \cap B) = P(A)P(B)$$