Addition Rule for Overlapping Events

The word or in probability usually means that event A happens, event B happens, or both happen. To find that probability, add the chance of A to the chance of B, then subtract the chance that both happen: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The union symbol ∪ means A or B; the intersection symbol ∩ means both A and B.

Why subtract the overlap?

Imagine a group of 30 students. Twelve are enrolled in art, 10 are enrolled in music, and 4 take both. Adding 12 and 10 gives 22, but the four students taking both courses have been counted twice. Subtract them once: 12 + 10 − 4 = 18 students take art or music. The probability of selecting one of those students at random is 18/30 = 3/5.

The same logic works with probabilities. If P(A) = 0.40, P(B) = 0.35, and P(A and B) = 0.10, then P(A or B) = 0.40 + 0.35 − 0.10 = 0.65. Check that the result is between 0 and 1. The probability of either event cannot exceed 1.

Mutually exclusive events

Two events are mutually exclusive when they cannot occur together in the same trial. A single roll of a standard die cannot show both a 2 and a 5, so P(2 or 5) = P(2) + P(5) = 1/6 + 1/6 = 1/3. Here P(A and B) = 0, so the general formula reduces to addition.

Do not assume events are mutually exclusive just because they have different names. In a card draw, 'a heart' and 'a king' overlap at the king of hearts. For a standard 52-card deck, P(heart or king) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13. Adding without subtracting the overlap would produce 17/52, which double-counts one card.

Use a table or Venn diagram to find the overlap

A two-way table often gives the counts needed for the formula. Find the total in event A, the total in event B, the number in both, and the overall sample size. Convert each count to a probability using the same total, then substitute. Alternatively, draw two overlapping circles: the shared region is A and B, while the full area covered by either circle is A or B. This picture makes the double count visible.

If a question gives only the separate event counts and not the overlap, look for enough information to calculate it. For example, if a class has 30 students, 12 study art, 10 study music, and 18 study at least one, then the overlap is 12 + 10 − 18 = 4. Rearranging the addition rule can recover a missing quantity.

Watch the wording

'A or B' includes both unless the problem explicitly says 'A or B, but not both.' Exclusive or means exactly one of the events occurs. Its probability is P(A) + P(B) − 2P(A and B), because the outcomes in the overlap should be removed entirely after being counted twice. A question may instead ask for 'at least one,' which uses the ordinary inclusive-or formula.

Compare addition with multiplication. The addition rule finds the chance that at least one of two events occurs in a single experiment or sample. Multiplication rules are used for both events occurring, with independence or conditional probability determining the correct product. Translate the words into the event requested before choosing a formula.

Common errors

• Adding event probabilities without subtracting a nonzero overlap.
• Subtracting the overlap twice when the question asks for inclusive or.
• Treating different events as mutually exclusive without checking whether both can occur.
• Using different denominators for counts from the same sample.
• Confusing 'A or B' with 'A and B.'
• Reporting a probability below 0 or above 1, which signals an arithmetic or setup error.

Recover a missing overlap from a survey

Suppose 40 people are asked whether they use a bus pass or a rail pass. Twenty-two use a bus pass, 17 use a rail pass, and 31 use at least one. The addition rule gives 31 = 22 + 17 − both, so both = 8. This rearrangement is useful when a problem gives the union and the separate event totals but hides the intersection. Check the answer against the margins: the overlap cannot exceed either event total, and it cannot be negative.

If a question gives a Venn diagram, add the regions carefully. The shared center belongs to both events, while the outer portions belong to one event each. To find 'A or B,' add the three regions inside either circle once. To find 'A and B,' use only the center. To find 'exactly one,' add the two nonoverlapping outer regions and leave out the center.

Use complements when they simplify the count

Sometimes it is easier to find the chance that neither event occurs and subtract from 1. Since 'at least one of A or B' is the complement of 'neither A nor B,' P(A or B) = 1 − P(neither A nor B). This approach is useful when a sample space lists many outcomes where neither event happens, but only a few where one or both do. The result should agree with the addition rule when both methods can be applied.

Do not use P(A or B) = P(A) + P(B) unless you know the overlap is zero. If events can happen together, the sum may exceed the actual union because shared outcomes have been counted twice. The general formula works in both cases and is the safest default.

Exam takeaway

For inclusive or, add the two event probabilities and subtract their intersection once. If the events cannot happen together, the intersection is zero and simple addition is enough. Check whether the problem means at least one event or exactly one, and use the table, counts, or diagram to identify any overlap.