Union and intersection of events
In probability, an event is a set of outcomes inside a sample space \(S\). With two events \(A\) and \(B\),
we often want to know the probability that at least one happens or that both happen.
These two ideas correspond to the union and intersection.
Key definitions
-
Union: \(A\cup B\) means “\(A\) or \(B\)” (inclusive OR). It includes outcomes in \(A\), in \(B\), or in both.
-
Intersection: \(A\cap B\) means “\(A\) and \(B\)”. It includes only outcomes that belong to both events.
-
Complement: \(A^c\) (or \(A'\)) means “not \(A\)”. It includes outcomes in \(S\) that are not in \(A\).
The addition rule (inclusion–exclusion for two events)
If you add \(P(A)\) and \(P(B)\), any outcomes in the overlap \(A\cap B\) get counted twice—once in \(P(A)\) and once in \(P(B)\).
The correction is to subtract the overlap once:
\[
P(A\cup B)=P(A)+P(B)-P(A\cap B).
\]
This formula is valid for any two events, whether they overlap a lot, a little, or not at all.
If you know \(P(A)\), \(P(B)\), and \(P(A\cap B)\), you can compute the union directly. If instead you know the union,
you can rearrange the formula to solve for the intersection:
\[
P(A\cap B)=P(A)+P(B)-P(A\cup B).
\]
Mutually exclusive vs. independent
Two events are mutually exclusive (disjoint) if they cannot happen together. Then \(A\cap B=\varnothing\),
so \(P(A\cap B)=0\), and the union simplifies to \(P(A\cup B)=P(A)+P(B)\).
This is different from independence, which means “one happening does not change the probability of the other.”
Independent events generally do overlap, and satisfy \(P(A\cap B)=P(A)\,P(B)\) (not zero unless one event has probability 0).
Exclusive-or and “neither”
Sometimes we want “exactly one of \(A\) or \(B\)” (exclusive-or). This removes the overlap:
\[
P(A\oplus B)=P(A)+P(B)-2P(A\cap B).
\]
Another common quantity is the probability that neither event happens, which is the complement of the union:
\[
P(\text{neither})=1-P(A\cup B).
\]
Venn diagrams and quick sanity checks
A Venn diagram is a visual model where the sample space is the outer region and events are drawn as shapes inside it.
The overlap represents \(A\cap B\) and the total covered region represents \(A\cup B\).
No matter how you compute, your numbers should respect:
\[
0\le P(A\cap B)\le \min(P(A),P(B)),\qquad \max(P(A),P(B))\le P(A\cup B)\le 1.
\]
If a set of inputs violates these inequalities, the event description is inconsistent.
