Start with the sample-space rectangle
A two-event probability Venn diagram commonly uses a rectangle for all possible outcomes and circles for events A and B. The overlap belongs to both events. The region outside both circles remains inside the sample space and represents neither. These four disjoint regions are A only, both, B only and neither. Their probabilities sum to one. The size of a printed circle is not necessarily drawn to scale, so use numerical labels rather than measuring its visual area.
Place an original set of counts
Suppose 100 equally weighted records contain 25 in A only, 15 in both, 20 in B only and 40 in neither. The complete A circle contains 25+15=40 records, and the complete B circle contains 20+15=35. Dividing by the shared total gives P(A)=0.4 and P(B)=0.35. A number written in the A-only region is not automatically the total probability of A. Identify whether labels represent whole events or exclusive regions before using them.
Read intersection and union
The intersection A∩B is the overlap, with probability 15/100=0.15. The union A∪B contains all three inside regions, so its probability is (25+15+20)/100=0.6. Adding P(A)+P(B) gives 0.75 and counts the overlap twice; subtracting 0.15 corrects it. Neither is 40/100=0.4, the complement of the union. These relationships follow from which outcomes belong to each event, not from a presumption that the circles are independent.
Read only and exactly-one language
A only means A∩B-complement, with probability 0.25 in the example. B only has probability 0.20. Exactly one combines those two disjoint side regions and gives 0.45. At least one also includes the overlap and gives 0.60. Watch the word only in problem statements: it changes the event being counted. A diagram can make the distinction visible, but the accompanying numerical labels still need a stated interpretation and common denominator.
| Region | Count | Probability |
|---|---|---|
| A only | 25 | 0.25 |
| Both | 15 | 0.15 |
| B only | 20 | 0.20 |
| Neither | 40 | 0.40 |
Whole A totals 40; whole B totals 35. Circle sizes need not be to scale.
Change denominators for conditional probability
P(A given B) restricts attention to the complete B circle. Of its 35 records, 15 also belong to A, so P(A|B)=15/35=3/7, approximately 0.428571429. P(B|A)=15/40=0.375 uses the A total instead. The overlap count stays 15 while the denominator changes. Conditional probabilities therefore need not match in the two directions. If the conditioning event has probability zero, the ordinary ratio is undefined, not a valid numerical zero.
Check that every region is nonnegative
If the whole-event probabilities are a and b and the overlap is j, the four regions are a−j, j, b−j and 1−a−b+j. Each must be at least zero. An overlap larger than either whole circle would make an only-region negative. An overlap too small when a+b>1 would make neither negative. Reject those combinations rather than drawing a misleading diagram or clipping individual values until the total happens to fit.
Do not infer independence from circle placement
Overlapping circles mean that a joint event is possible in the diagram's model, not that its probability equals the product of marginals. In this example, 0.4×0.35=0.14 differs from the supplied overlap 0.15, so the numbers are not exactly independent. Circles drawn apart represent exclusivity if their disjointness is part of the stated event definition. Their appearance alone cannot establish a statistical relationship when a drawing was merely illustrative or not numerically specified.
Reproduce the partition in the calculator
Use known-overlap mode with P(A)=0.4, P(B)=0.35 and overlap 0.15. Confirm only-A 0.25, only-B 0.20, neither 0.40 and exactly-one 0.45. The Probability Calculator reports numerical regions rather than editing a Venn illustration. Its conditional input, when that mode is selected, is B given A; to recreate this partition supply 0.375, not 3/7. Check the direction label rather than assuming the tool accepts whichever conditional probability a question asks for.
Preserve counts, weights and context
A simple count-to-probability conversion assumes the records share one relevant total and are equally weighted for the question. Survey weights or unequal outcome probabilities need appropriate weighted totals instead. Keep the event definitions and time window with the diagram so a record cannot move between regions merely because the definition changed. A valid partition can explain association but does not establish that one event causes the other. Record the four disjoint values as a compact check before using any derived union or conditional ratio.