Introduction to Combined Events

Definition of Intersection: The intersection of two events, $A$ and $B$, is the set of outcomes that are included in both event $A$ and event $B$. It represents the logical 'AND' condition where both criteria must be satisfied simultaneously.

Notation: The intersection is denoted by the symbol $\cap$, written as $A \cap B$. This notation is commutative, meaning that $A \cap B$ is identical to $B \cap A$ because the order of the events does not change the shared outcomes.

Visual Identification: In a Venn diagram, the intersection is the overlapping region between the two circles. In a two-way table, it is the specific cell where the row for one event meets the column for the other.

Definition of Union: The union of two events, $A$ and $B$, is the set of outcomes that are included in event $A$, event $B$, or both. It represents the logical 'OR' condition, which is inclusive in probability theory.

Notation: The union is denoted by the symbol $\cup$, written as $A \cup B$. Like the intersection, the union is commutative, so $A \cup B$ is the same as $B \cup A$.

Scope of Outcomes: A union includes every outcome that belongs to at least one of the specified events. This means the union is always at least as large as the largest individual event involved.

Complement of a Union: The event $(A \cup B)'$ represents the outcomes that are in neither $A$ nor $B$. This is logically equivalent to the intersection of the complements, $A' \cap B'$, which consists of outcomes that are not in $A$ AND not in $B$.

Probability Summation: Because an event and its complement are mutually exclusive and exhaustive, the probability of a union and the probability of its complement must add up to exactly $1$. This is expressed as $P(A \cup B) + P((A \cup B)') = 1$.

De Morgan's Logic: These relationships allow statisticians to calculate the probability of 'at least one' event occurring by instead finding the probability that 'none' of the events occur and subtracting that value from $1$.

The 'At Least' Rule: When an exam question asks for the probability of 'at least one' event occurring, it is almost always a signal to calculate the union $P(A \cup B)$. If the union is difficult to calculate directly, use the complement: $1 - P(\text{neither})$.

The 'Neither' Logic: If you are asked to find the probability that neither event $A$ nor event $B$ occurs, you are looking for $P(A' \cap B')$. This is equivalent to $1 - P(A \cup B)$, which is often easier to find if you have a two-way table.

Sanity Checks: Always verify that your calculated probabilities for intersections are less than or equal to the probabilities of the individual events. Conversely, the probability of a union must be greater than or equal to the probability of any single event involved.