Does the addition rule require independence?

No. The general formula holds whether or not the events are independent, provided the intersection probability is correctly specified.

What if the events cannot happen together?

Their intersection probability is zero, so the formula reduces to P(A) + P(B).

What can be said if only the individual probabilities are known?

The union is at least the larger individual probability and at most the smaller of one and their sum. More information about overlap is needed for an exact value.

AI Glossary / Statistics and probability / Beginner

Additive law of probability

Definition

The additive law of probability states that the probability of one of several mutually exclusive events occurring is the sum of their individual probabilities. For two events that can overlap, the general addition rule subtracts their intersection: P(A or B) = P(A) + P(B) - P(A and B).

What it means

The rule concerns an inclusive "or": A happens, B happens, or both happen. When events are mutually exclusive, both cannot happen in the same defined trial, so their intersection has probability zero and simple addition is valid.

The source glossary uses "additive law" for the mutually exclusive case. The general rule is included here because real categories frequently overlap. Adding their probabilities without removing the overlap counts shared outcomes twice. This is a bookkeeping principle about events, not an assumption that the events are independent.

How it works

Define the random experiment and events clearly. If the events are disjoint, add their probabilities. If they overlap, identify the probability that both occur and subtract it once from the sum. The result must lie between zero and one and cannot be smaller than either individual event probability.

For more than two overlapping events, inclusion-exclusion adds and subtracts intersections of different orders. For pairwise disjoint events, the calculation remains a simple sum. A set of disjoint events covering every possible outcome forms a partition, and its probabilities sum to one.

Use probabilities referring to the same population, period and conditions. The analogous conditional rule also works, but every term must be conditioned on the same event. Mixing an overall percentage with a subgroup percentage generally does not produce a meaningful union probability.

Three practical examples

1. A service team combines mutually exclusive ticket-status categories to calculate the probability that a randomly chosen ticket is either waiting or actively being handled.

2. A product analyst combines mobile and desktop usage while subtracting customers who used both, avoiding a double-counted audience estimate.

3. A reliability analyst calculates the probability of at least one of two specified issues, accounting for the possibility that both occur together.

Think of it this way

Think of combining two guest lists. Adding their lengths works only if no person appears on both. If some guests are shared, subtract the shared count once to obtain the number of distinct people. The probability rule performs the same correction on proportions rather than raw list lengths.

Worked example

Consider an original dataset of 200 customers. Eighty used mobile access during a stated week, 70 used desktop access and 30 used both. The number who used at least one is 80 + 70 - 30 = 120, so the probability for a uniformly selected customer is 120/200 = 0.60.

The equivalent probability calculation is 0.40 + 0.35 - 0.15 = 0.60. Exactly one mode was used by (80 - 30) + (70 - 30) = 90 customers, giving 0.45. Neither mode was used by 200 - 120 = 80 customers, giving 0.40.

For contrast, suppose each ticket has exactly one status: 20% waiting, 50% active and 30% closed. Waiting and active are mutually exclusive, so their union is simply 0.20 + 0.50 = 0.70. The customer modes and ticket statuses use different overlap structures, which explains why their calculations differ.

Illustrative case study

Fictional case study: Reachwise reports how many customers used at least one of two support channels. Its first dashboard adds channel percentages and produces a total above 100% because many customers used both.

The analysts align the reporting period and customer identifier, calculate the shared group and apply the addition rule. They then label the result "used at least one channel" and report "used exactly one" separately. The dashboard also explains whether the denominator includes all eligible customers or only customers who sought support.

This fictional case illustrates event definition and deduplication. It does not establish the quality of the identity matching or imply that channel use measures satisfaction or support effectiveness.

Common misconceptions and mistakes

Mutually exclusive and independent are different concepts. Two events with positive probabilities cannot be both mutually exclusive and independent: exclusivity makes their joint probability zero, while independence would make it the positive product of their probabilities.

Another mistake is treating "or" as exactly one when the formula means at least one. Also, adding percentages from different denominators is invalid without a suitable reformulation. When overlap is unknown, the exact union is generally unknown; simple addition supplies only an upper bound, capped at one, rather than an exact answer.

Business use

The additive law supports consistent reporting of audiences, incidents, categories and outcomes. It is especially valuable when teams combine metrics from systems that allow multiple labels per person or event.

Define the unit being counted and ensure that intersections are measured on that same unit. A customer appearing in two channels is different from two separate contacts, and either may be the appropriate business measure. Reconcile identifiers and time windows before calculating overlap. When reporting estimated probabilities, retain uncertainty and avoid presenting an arithmetic combination as more precise than its underlying measurements justify.

Check your understanding

P(A) = 0.3, P(B) = 0.5 and P(A and B) = 0.2. What is the probability that at least one occurs?

Answer: 0.3 + 0.5 - 0.2 = 0.6. The overlap is subtracted once because it was counted in both individual probabilities.

Related terms

Mutually Exclusive Events
Independence
union
intersection
inclusion-exclusion
complement
conditional probability and partition

Learn next

draw a two-event diagram and label the A-only, B-only, shared and neither regions. Verify that the regions sum to one. Then compare the probabilities of at least one, exactly one and both, using the same definitions throughout.

Sources

Primary sources: Howell, Statistical Methods for Psychology glossary, Additive law of probability: https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html

Penn State, STAT 500, Probability: https://online.stat.psu.edu/stat500/Lesson02

OpenStax, Contemporary Mathematics, The Addition Rule for Probability: https://openstax.org/books/contemporary-mathematics/pages/7-8-the-addition-rule-for-probability

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