Ask two different questions about the events
Independent events do not change one another's probabilities when the relevant conditioning probabilities are defined. Mutually exclusive events cannot occur together. These statements answer different questions: whether information changes a probability, and whether there is any shared outcome. Two events with positive probability cannot be both independent and mutually exclusive. Distinguish the concepts before choosing a multiplication or addition rule, rather than treating both terms as a general description of unrelated events.
Start by defining A and B within one sample space. This guide uses a hypothetical complete collection of 200 records and considers selecting one record uniformly. Event A is membership in one group and event B membership in another. Their counts and shared membership define exact probabilities for this constructed collection. The example is not a sample-based estimate of a larger population or evidence about how real people behave.
Read a coherent overlap table before calculating
Suppose 20 records belong to both groups, 70 belong only to A, 60 only to B and 50 to neither. The four disjoint counts total 200. Group A contains 20 + 70 = 90 records, so P(A) = 90 / 200 = 0.45. Group B contains 20 + 60 = 80 records, so P(B) = 80 / 200 = 0.4. The overlap is P(A ∩ B) = 20 / 200 = 0.1.
The Probability Calculator's known-overlap mode represents this table: enter 0.45, 0.4 and 0.1 on the decimal scale. If using percentages, enter 45, 40 and 10 instead. Changing the input scale loads example values, so replace them with the intended data afterward. An entry of 45 in decimal mode is invalid; it is not automatically interpreted as 45%. Keep the scale and event definitions in the saved calculation record.
| Disjoint region | Records | Probability |
|---|---|---|
| Both A and B | 20 | 20 / 200 = 0.10 |
| A only | 70 | 70 / 200 = 0.35 |
| B only | 60 | 60 / 200 = 0.30 |
| Neither | 50 | 50 / 200 = 0.25 |
Counts total 200 and probabilities total 1. Marginal A includes both plus A only; marginal B includes both plus B only.
Use the overlap to calculate inclusive OR and exactly one
Inclusive OR asks whether A, B or both occur. Add the two marginals and subtract the shared region once: P(A ∪ B) = 0.45 + 0.4 − 0.1 = 0.75. There are 150 qualifying records, matching 70 + 60 + 20. Neither event has probability 50 / 200 = 0.25. Adding marginals without subtracting overlap would count the 20 shared records twice and incorrectly report 0.85 for this collection.
Exactly one excludes the shared region altogether. Its probability is 70 / 200 + 60 / 200 = 0.65, or equivalently P(A) + P(B) − 2P(A ∩ B). This differs from the 0.75 inclusive union. Words such as either can be ambiguous in an informal question, so identify whether both should qualify before entering the model. The interface labels at least one and exactly one separately to preserve that distinction.
Compare the table with an explicit independence model
Independence would require P(A ∩ B) = P(A)P(B). With marginals 0.45 and 0.4, that product is 0.18, equivalent to 36 shared records in a collection of 200. Our defined table instead has 20 shared records and overlap 0.1, so these exact events are not independent. Choosing independent mode would answer a different hypothetical model that shares the same marginals but has a different joint distribution.
This exact constructed comparison does not provide a general significance test for observed sample counts. In a real sample, differences from an expected product can arise through sampling variation or data problems, and independence assessment needs an appropriate design and method. The calculator checks arithmetic under the relationship you select; it does not establish independence from two marginal probabilities or prove dependence in a wider population from a small observed discrepancy.
Recognize what mutually exclusive would change
Mutual exclusivity sets overlap to zero. If a different coherent model retained P(A) = 0.45 and P(B) = 0.4 while making the groups disjoint, its union would be 0.85, neither would be 0.15 and exactly one would also be 0.85. That model cannot represent the original table, which explicitly contains 20 shared records. A relationship is part of the question, not a menu choice made to obtain a preferred numerical answer.
Two positive-probability independent events have a positive product, whereas mutually exclusive events have zero overlap. At least one event must have probability zero for both properties to hold simultaneously. Also check the sum when selecting exclusivity: probabilities 0.75 and 0.4 cannot be disjoint because their union would exceed one. An inconsistency message means the inputs or relationship need correction, not that an impossible answer should be capped at 100%.
Check the conditioning direction and denominator
In the original collection, P(B | A) = 20 / 90 = 2 / 9, approximately 0.222222222222. Knowing A restricts the reference set to its 90 records. In the reverse direction, P(A | B) = 20 / 80 = 0.25. The overlap count is unchanged, but the denominator changes. Both conditional probabilities are therefore meaningful and different. A phrase containing given identifies which event belongs in that denominator.
Known-conditional mode specifically requests B given A. Entering the appropriately precise decimal for 2 / 9 with P(A) = 0.45 recovers overlap approximately 0.1; the fraction itself is not an accepted input expression. If the conditioning event has probability zero, the elementary ratio is undefined rather than zero. Do not insert a fabricated conditional value to bypass that limit. Known-overlap mode can still represent a mathematically established zero overlap without claiming an undefined ratio exists.
Report bounds when the relationship is unknown
Suppose only the marginals 0.45 and 0.4 are available and the group table has been lost. The overlap can range from max(0, 0.45 + 0.4 − 1) = 0 to min(0.45, 0.4) = 0.4. The union can range from 0.45 to 0.85. Neither the 0.18 independence overlap nor the original 0.1 overlap is determined by those two inputs alone. Unknown-relationship mode reports the bounds instead of choosing between them.
Once a joint or conditional value is supplied, it must fit those same constraints. An overlap of 0.5 is impossible because event B itself has probability only 0.4. Validation catches that contradiction, while relative numerical checks allow ordinary rounding residuals at legitimate boundaries. Very small products that underflow available precision are rejected rather than displayed as genuine zeros. Decimal output still has finite precision and is not an exact-rational probability proof.
Match the calculation to the information available
Keep the event definitions, common sample space, input scale and source of the relationship alongside the result. For the original table, a complete record gives marginals 0.45 and 0.4, overlap 0.1, union 0.75 and neither 0.25. The four disjoint regions reconstruct the total probability of one. Saving that partition is more informative than recording an unlabeled 75% whose meaning could later be mistaken for both or exactly one.
Use the sampling guide when the practical question concerns drawing records with or without replacement; the mechanism can alter dependence and needs its own model. This educational two-event workflow does not predict outcomes, estimate missing probabilities or assess risk from unverified assumptions. First identify what is known, then use the corresponding rule. If the relationship remains unknown, a justified interval is a more complete answer than an unjustified independence calculation.