The Addition Rule

The addition rule finds the probability that one event or another happens. You add the two probabilities, then subtract the part they share so you do not count it twice.

In symbols, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ . The subtraction is the whole point — the overlap gets counted once in $P(A)$ and again in $P(B)$ , so you take one copy back out.

Why you subtract the overlap

Imagine two overlapping circles in a Venn diagram. Adding $P(A)$ and $P(B)$ counts the shaded overlap region twice — once for each circle.

Subtracting $P(A \cap B)$ removes that double count, leaving each outcome counted exactly once. That is the entire logic of the rule.

The figure is that Venn diagram: circle $A$ and circle $B$ overlap in the lens-shaped region $A \cap B$ .

Mutually exclusive events

When two events cannot both happen — rolling a $2$ and rolling a $5$ on one die — their overlap is zero. Then $P(A \cap B) = 0$ and the rule simplifies to $P(A \cup B) = P(A) + P(B)$ .

So mutually exclusive is just the special case where there is nothing to subtract. Always check for overlap before you decide you can skip the subtraction.

Worked examples

Example 1: with overlap

$P(A) = 0.6$ , $P(B) = 0.3$ , and $P(A \cap B) = 0.1$ . Find $P(A \cup B)$ .

Addition rule

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Substitute

0.6 + 0.3 - 0.1

Simplify

0.8

Answer: $0.8$

Example 2: mutually exclusive

On one die roll, find P(rolling a 2 or a 5).

They cannot both happen, so no overlap

P(A \cap B) = 0

Add the probabilities

\tfrac{1}{6} + \tfrac{1}{6}

Simplify

\tfrac{2}{6} = \tfrac{1}{3}

Answer: $\tfrac{1}{3}$

Try one yourself

Common questions

When can I skip the subtraction?

Only when the events are mutually exclusive — they cannot both occur, so their overlap probability is zero. Otherwise you must subtract $P(A \cap B)$ .

What does the union symbol mean?

$A \cup B$ means 'A or B or both' — the event that at least one of them happens. $A \cap B$ means 'A and B together' — the overlap.

Can the addition rule ever give more than 1?

No. If your answer exceeds 1, you forgot to subtract the overlap. A correct probability always lands between 0 and 1.

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1

The Addition Rule

Why you subtract the overlap

Mutually exclusive events

Worked examples

Example 1: with overlap

Example 2: mutually exclusive

Try one yourself

Common questions

Keep learning

Two-Way Frequency Tables

Sample Spaces & Basic Probability

Conditional Probability & Independence