Probability of A or B When Events Overlap: Subtract the Double Count

For an inclusive “A or B,” add the two probabilities and subtract their overlap once: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The word “or” includes both.

Why subtraction is necessary

Imagine placing every member of A in one list and every member of B in another. Anyone belonging to both appears twice in the combined lists. Subtracting one copy of the overlap leaves that person counted once, as required by the union. This is a counting argument, not an assumption that the events are independent.

If the events cannot happen together, the overlap is zero and the formula reduces to simple addition. If they are independent, their overlap can be calculated by multiplication. Overlap and independence are separate questions; one label does not replace the other.

Worked example: one integer chosen uniformly

Choose one integer from 1 through 30, with each integer equally likely. Let A mean “divisible by 3,” and B mean “divisible by 5.” There are 10 multiples of 3 and 6 multiples of 5. The numbers 15 and 30 belong to both groups.

The union therefore contains 10 + 6 − 2 = 14 integers. Its probability is 14/30 = 7/15. Equivalently, P(A) = 1/3, P(B) = 1/5, and P(A ∩ B) = 1/15. Adding 1/3 + 1/5 − 1/15 gives 7/15.

A direct list offers a check: 3, 5, 6, 9, 10, 12, 15, 18, 20, 21, 24, 25, 27, 30. Notice that “or” includes 15 and 30. If a question says “exactly one of A and B,” remove the overlap entirely: 10 + 6 − 2 × 2 = 12, giving probability 2/5.

Useful boundary checks

• The union must be at least as likely as either event alone.
• The union cannot exceed 1.
• The overlap cannot exceed either event’s probability.
• All probabilities must describe the same experiment and sample space.

These checks can expose incompatible data. For example, if P(A) = 0.8 and P(B) = 0.7, an overlap of 0.1 would produce union 1.4. The supplied probabilities cannot all hold together.

Common mistake

Do not substitute P(A)P(B) for the overlap unless independence is given or established. In surveys, the overlap may be supplied directly as a count. Use that count with the same total as the other groups, and distinguish “neither,” “at least one,” and “exactly one” before calculating.

Dono lists mein jo common hai, usko ek baar minus karo.

Try three questions, then check the working

1. P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.2. Find the union.

   Use 0.5 + 0.4 − 0.2 = 0.7. The overlap was counted twice during addition, so one copy is removed.

2. Among 50 students, 28 like chess, 20 like music, and 10 like both. How many like neither?

   The union contains 28 + 20 − 10 = 38 students. Neither contains 50 − 38 = 12 students.

3. From 1 to 12, find the probability of an even number or a multiple of 3.

   There are 6 evens, 4 multiples of 3, and 2 multiples of 6. Union count = 6 + 4 − 2 = 8; probability = 8/12 = 2/3.

Build the next step

• Probability Class 10 Notes: Formula, Sample Space, Coins, Dice and Cards
• Probability Class 12 Notes: Conditional Probability, Bayes’ Theorem and Distributions

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Reference and authorship

The examples and practice questions here are original, with calculations checked separately. Definitions and related background: OpenStax: Two Basic Rules of Probability. Prepared with AI assistance and editorial arithmetic review; no textbook affiliation is claimed.