Set theory provides the structural language used to combine and manipulate probability events.
The complement of an event A represents the logical negation of A, comprising all sample space outcomes not in A.
Since an event and its complement partition the sample space into mutually exclusive parts, their probabilities sum to one.
The subtraction rule states that the probability of event A equals one minus the probability of its complement.
This rule is particularly effective for evaluating at least one problems in multi-step experiments.
For example, computing the probability of getting at least one six in four dice rolls is simpler by evaluating no sixes first.
The probability of rolling no sixes across four independent rolls is five-sixths raised to the fourth power.
Subtracting that value from one immediately yields the probability of rolling at least one six.
The union of events A and B represents the event that A occurs, B occurs, or both occur simultaneously.
The general addition rule calculates the union as the probability of A plus the probability of B minus the intersection of A and B.
Subtracting the intersection is necessary because outcomes belonging to both sets are counted twice when adding raw probabilities.
If two events are mutually exclusive, their intersection is empty, and the probability of their union simplifies to simple addition.
Mutually exclusive events cannot happen together: flipping a coin once cannot yield both heads and tails simultaneously.
A frequent beginner trap is confusing mutually exclusive events with statistically independent events.
Mutually exclusive events with non-zero probability can never be independent, because the occurrence of one prevents the other.