Addition Rule

Q: Why do you subtract P(A and B) in the Addition Rule?

When you add $P(A)$ and $P(B)$, any outcomes that belong to both events get counted twice — once in $P(A)$ and once in $P(B)$. Subtracting $P(A \text{ and } B)$ removes this double-counting so each outcome is counted exactly once. This is the same principle as the inclusion-exclusion formula in set theory.

Addition Rule
Sum Rule for Probability

A method for finding the probability that either or both of two events occurs.

Addition Rule:

If events A and B are mutually exclusive (disjoint), then

P(A or B) = P(A) + P(B)

Otherwise,

P(A or B) = P(A) + P(B) – P(A and B)

Example 1:
mutually exclusive

In a group of 101 students 30 are freshmen and 41 are sophomores. Find the probability that a student picked from this group at random is either a freshman or sophomore.

Note that P(freshman) = 30/101 and P(sophomore) = 41/101. Thus

P(freshman or sophomore) = 30/101 + 41/101 = 71/101

This makes sense since 71 of the 101 students are freshmen or sophomores.

Example 2:
not mutually exclusive

In a group of 101 students 40 are juniors, 50 are female, and 22 are female juniors. Find the probability that a student picked from this group at random is either a junior or female.

Note that P(junior) = 40/101 and P(female) = 50/101, and P(junior and female) = 22/101. Thus

P(junior or female) = 40/101 + 50/101 – 22/101 = 68/101

This makes sense since 68 of the 101 students are juniors or female.

Not sure why? When we add 40 juniors to 50 females and get a total of 90, we have overcounted. The 22 female juniors were counted twice; 90 minus 22 makes 68 students who are juniors or female.

See also

Multiplication rule

Key Formula

P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

Where:

$P(A)$ = Probability that event A occurs
$P(B)$ = Probability that event B occurs
$P(A \text{ and } B)$ = Probability that both events A and B occur simultaneously (the overlap)
$P(A \text{ or } B)$ = Probability that at least one of the two events occurs

Worked Example

Problem: A standard deck of 52 cards contains 13 hearts and 4 kings (one of which is the king of hearts). You draw one card at random. What is the probability that the card is a heart or a king?

Step 1: Identify the individual probabilities. There are 13 hearts and 4 kings in the deck.

P(\text{heart}) = \frac{13}{52}, \quad P(\text{king}) = \frac{4}{52}

Step 2: Determine the overlap. The king of hearts is both a heart and a king, so the events are not mutually exclusive.

P(\text{heart and king}) = \frac{1}{52}

Step 3: Apply the Addition Rule by adding the individual probabilities and subtracting the overlap.

P(\text{heart or king}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52}

Step 4: Simplify the result.

P(\text{heart or king}) = \frac{16}{52} = \frac{4}{13}

Answer: The probability of drawing a heart or a king is

\frac{4}{13} \approx 0.308

Another Example

This example uses a single die rather than a deck of cards, and includes a verification step where all favorable outcomes are listed directly to confirm the Addition Rule gives the correct answer.

Problem: You roll a single six-sided die. What is the probability that you roll a number less than 3 or an even number?

Step 1: List the outcomes for each event. 'Less than 3' gives {1, 2}. 'Even' gives {2, 4, 6}.

P(\text{less than 3}) = \frac{2}{6}, \quad P(\text{even}) = \frac{3}{6}

Step 2: Find the overlap. The number 2 satisfies both conditions, so the events share one outcome.

P(\text{less than 3 and even}) = \frac{1}{6}

Step 3: Apply the Addition Rule.

P(\text{less than 3 or even}) = \frac{2}{6} + \frac{3}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}

Step 4: Verify by listing. The outcomes that satisfy at least one condition are {1, 2, 4, 6} — that is 4 out of 6.

\frac{4}{6} = \frac{2}{3} \checkmark

Answer: The probability is

\frac{2}{3} \approx 0.667

Frequently Asked Questions

When do you use the Addition Rule vs. the Multiplication Rule?

Use the Addition Rule when you want the probability of event A OR event B (at least one happening). Use the Multiplication Rule when you want the probability of event A AND event B (both happening). The key word 'or' signals the Addition Rule, while 'and' signals the Multiplication Rule.

What happens to the Addition Rule when events are mutually exclusive?

When events are mutually exclusive, they cannot happen at the same time, so

P(A \text{ and } B) = 0

. The formula simplifies to

P(A \text{ or } B) = P(A) + P(B)

. You can simply add the probabilities without subtracting anything.

Why do you subtract P(A and B) in the Addition Rule?

When you add

P(A)

and

P(B)

, any outcomes that belong to both events get counted twice — once in

P(A)

and once in

P(B)

. Subtracting

P(A \text{ and } B)

removes this double-counting so each outcome is counted exactly once. This is the same principle as the inclusion-exclusion formula in set theory.

Addition Rule vs. Multiplication Rule

	Addition Rule	Multiplication Rule
Key word	"or" — at least one event occurs	"and" — both events occur
Formula	P(A or B) = P(A) + P(B) − P(A and B)	P(A and B) = P(A) × P(B \| A)
Simplified case	If mutually exclusive: P(A or B) = P(A) + P(B)	If independent: P(A and B) = P(A) × P(B)
What it finds	Probability of the union of two events	Probability of the intersection of two events

Why It Matters

The Addition Rule appears in nearly every introductory probability and statistics course, from AP Statistics to college-level courses. You use it any time a problem asks for the chance of "at least one" of two events happening — situations like drawing certain cards, rolling dice outcomes, or analyzing survey data with overlapping categories. Mastering this rule also builds the foundation for more advanced topics like the inclusion-exclusion principle and Bayes' theorem.

Common Mistakes

Mistake: Forgetting to subtract the overlap when events are not mutually exclusive.

Correction: Always check whether the two events can occur at the same time. If they can, you must subtract P(A and B) to avoid double-counting the outcomes that satisfy both events.

Mistake: Assuming events are mutually exclusive when they are not.

Correction: Read the problem carefully. For example, being 'female' and being a 'junior' are not mutually exclusive because a student can be both. If in doubt, use the general formula — it works in all cases, since P(A and B) = 0 for mutually exclusive events anyway.

Related Terms

Probability — The broader field the Addition Rule belongs to
Event — The outcomes whose probabilities are combined
Mutually Exclusive — Special case where the overlap term is zero
Multiplication Rule — Finds P(A and B) instead of P(A or B)
Complement — Used with Addition Rule for 'at least one' problems
Conditional Probability — Probability of one event given another has occurred
Independent Events — Simplifies the Multiplication Rule, often used alongside
Sample Space — The set of all possible outcomes from which events are drawn