Fundamentals of Business Mathematics and Statistics · Probability

Addition Theorem of Probability: P(A or B) Explained

Updated 10 October 2026 · Fact-checked

The addition theorem gives the probability that at least one of two events happens. For any events A and B, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0, so you simply add P(A) and P(B).

Understand Addition Theorem of Probability

Probability measures how likely an event is, on a scale from 0 to 1. Often a question asks for the chance that A or B happens. In set language this is the union, A ∪ B. It means at least one of the two events occurs. Both occurring also counts.

Start with mutually exclusive events. These cannot happen together. On one throw of a die, getting 2 and getting 5 are mutually exclusive. The outcomes do not overlap, so you add the probabilities: P(A ∪ B) = P(A) + P(B).

Now take events that can happen together, called non-exclusive events. On one throw of a die, "even number" and "number greater than 3" both include 4 and 6. If you add P(A) and P(B), the overlap is counted twice. So you subtract it once: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The general formula covers both cases. For mutually exclusive events the last term is zero.

The complement of A, written A′, means A does not happen. P(A′) = 1 − P(A). This helps with "at least one" questions. The probability of at least one of A or B is P(A ∪ B). Its complement is that neither happens, so P(A ∪ B) = 1 − P(neither A nor B).

For three events, the rule extends with an inclusion-exclusion pattern. At Foundation level, two-event problems are far more common.

Key formulas to remember

Addition theorem (general): P(A ∪ B) = P(A) + P(B) − P(A ∩ B); Works for any two events A and B.
Mutually exclusive events: P(A ∪ B) = P(A) + P(B); Use only when A and B cannot occur together, so P(A ∩ B) = 0.
Complementary event: P(A′) = 1 − P(A); A and A′ are mutually exclusive, and P(A) + P(A′) = 1.
Neither A nor B: P(A′ ∩ B′) = 1 − P(A ∪ B); This is De Morgan's law applied to probability.
At least one of A, B: P(at least one) = P(A ∪ B) = 1 − P(A′ ∩ B′); Use the complement when it is easier to find P(neither).
Three events: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(B ∩ C) − P(A ∩ C) + P(A ∩ B ∩ C); Add singles, subtract pairs, add back the triple overlap.

How to solve Addition Theorem of Probability questions

Use this method for any addition theorem question. The key is to decide whether the events overlap.

1Read the question and mark the words "or", "either", "at least one". These signal a union.
2Name the events clearly as A and B, and write down the given probabilities.
3Decide if the events are mutually exclusive. If they cannot happen together, P(A ∩ B) = 0.
4If they can overlap, find P(A ∩ B). It may be given, or you may count favourable outcomes common to both.
5Put the values into P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
6For "neither" or "none", subtract the union from 1.
7Check that your answer lies between 0 and 1 and is at least as large as the bigger of P(A) and P(B).
8Match the answer to the options and mark it.

Quickest way: Count the outcomes directly

When to use it: Use this for cards, dice, coins and numbered tickets, where the sample space is small and countable.

Write the total number of outcomes, for example 52 cards or 6 faces.
List or count outcomes in A or B, counting any common outcome only once.
Divide by the total. This gives the union directly and avoids formula errors.
If the question says "at least one" and the direct count looks long, find the count of outcomes in neither and subtract from the total.
Use the formula only when probabilities are given, not outcomes.

Common mistakes in Addition Theorem of Probability

Adding P(A) and P(B) without subtracting the overlap.
Students memorise the simple addition rule and forget it holds only for mutually exclusive events.
Fix: Ask first: can both happen together? If yes, subtract P(A ∩ B).
Treating mutually exclusive events as independent.
Both ideas sound like "separate" events.
Fix: Mutually exclusive means P(A ∩ B) = 0. Independent means P(A ∩ B) = P(A) × P(B). For events with non-zero probability, they cannot both be true.
Using P(A ∩ B) = P(A) × P(B) in every problem.
Students assume independence without being told.
Fix: Multiply only if the question states the events are independent. Otherwise use the given intersection or count outcomes.
Counting a common outcome twice in card or dice problems.
The overlap is easy to miss, such as the king of hearts in "king or heart".
Fix: List the overlap and subtract it once. For king or heart, 4 + 13 − 1 = 16 outcomes.
Getting an answer above 1 or below the larger single probability.
Students skip the final reasonableness check.
Fix: A union probability can never exceed 1 and is never smaller than P(A) or P(B). If it is, recheck the overlap.

Worked examples

Example 1

A card is drawn at random from a well-shuffled pack of 52 cards. Find the probability that it is a king or a heart.

Show the solution

Let A = king and B = heart.
P(A) = 4/52 and P(B) = 13/52.
The king of hearts is both, so P(A ∩ B) = 1/52.
P(A ∪ B) = 4/52 + 13/52 − 1/52 = 16/52.
Simplify: 16/52 = 4/13.

Answer: 4/13

Example 2

For two events A and B, P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2. Find the probability that neither A nor B occurs.

Show the solution

Use the addition theorem: P(A ∪ B) = 0.5 + 0.4 − 0.2 = 0.7.
Neither A nor B is the complement of the union.
P(neither) = 1 − 0.7 = 0.3.

Answer: 0.3

Exam tips

Look for the words "or", "either" and "at least one". They tell you to use the addition theorem.
Check the wording "mutually exclusive" before choosing the formula. If it is absent, assume the overlap may exist.
In "at least one" and "none" questions, compute the union first, then take 1 minus it where needed.
For card and dice questions, count outcomes. It is faster than using the formula and avoids errors.
There is no negative marking, so eliminate options above 1 or below the larger single probability, then answer every question.

Practice questions from Probability

Addition Theorem of Probability in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CA FoundationAddition Theorem of Probability

Addition Theorem of Probability: frequently asked questions

What is the addition theorem of probability?

It gives the probability that at least one of two events occurs. The rule is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The subtraction removes the overlap that was counted twice.

When can I simply add P(A) and P(B)?

Add them only when the events are mutually exclusive, meaning they cannot occur together. Then P(A ∩ B) = 0. If the events can overlap, you must subtract the common part.

How do I find the probability of at least one event?

For two events, it is P(A ∪ B). Often the quickest route is 1 − P(neither). Find the probability that none of the events happens, then subtract it from 1.

Are mutually exclusive events the same as complementary events?

No. Complementary events are mutually exclusive and together cover all outcomes, so their probabilities sum to 1. Mutually exclusive events cannot occur together, but their probabilities may sum to less than 1.