Venn Diagrams for Disjoint and Overlapping Events

Use the Regions to Organize Probabilities

A Venn diagram gives a picture of how two events relate to one another. In Drawing a Venn Diagram of Two Events, you learned that the overlap is \(A\cap B\), the outcomes in both events. Here, we will use the diagram as a map for placing given probabilities and finding missing ones. The previous tutorial, Mutually Exclusive Versus Independent Events, also emphasized that disjointness concerns whether events can happen together.

For two events \(A\) and \(B\), a diagram has four regions: outcomes in \(A\) but not \(B\), outcomes in both, outcomes in \(B\) but not \(A\), and outcomes in neither. Each region represents a probability. Because these regions do not overlap and together cover the entire sample space, their probabilities add to 1.

Definition: In a two-event Venn diagram, the four regions are A only, \(A\cap B\), B only, and neither. The probability of each event includes its exclusive region and the overlap, while the probability outside both circles is the probability of neither event.

To draw the diagram, first draw the sample-space boundary, then draw one circle for each event. If the events can overlap, let the circles intersect. If they are disjoint, draw separate circles. Write the intersection probability in the overlap first. Then use the event probabilities to fill in the exclusive regions. The space outside both circles is neither.

Diagram region	What it means	How to find its probability
A only	\(A\) occurs, but \(B\) does not	\(P(A)-P(A\cap B)\)
Both	\(A\) and \(B\) occur	\(P(A\cap B)\)
B only	\(B\) occurs, but \(A\) does not	\(P(B)-P(A\cap B)\)
Neither	Neither \(A\) nor \(B\) occurs	\(1-P(A\cup B)\)

Filling a Diagram for Disjoint Events

If \(A\) and \(B\) are disjoint, the circles have no shared outcomes. The overlap region is empty, so its probability is zero. The probability of \(A\) goes entirely in the A-only region, and the probability of \(B\) goes entirely in the B-only region. The probability of neither is whatever remains outside the circles.

Key rule: For disjoint events, \(P(A\cap B)=0\). Put \(P(A)\) in A only and \(P(B)\) in B only. Then find neither by subtracting their total from 1.

This is the visual version of the addition rule for mutually exclusive events: the probability of \(A\) or \(B\) is the sum of their probabilities. The diagram makes clear why there is no overlap to subtract. The result must be at most 1; otherwise, the supplied probabilities cannot describe disjoint events in the same probability model.

Worked Example: Filling a Disjoint-Events Diagram

In a made-up community survey, let \(A\) mean that a randomly selected resident uses a bicycle to commute, and let \(B\) mean that the resident uses a city bus to commute. For this question, each resident is counted in only one commute category. Suppose \(P(A)=0.28\) and \(P(B)=0.37\). Fill the Venn diagram regions.

State: The two commute events are disjoint under the stated categories. We need the probability in each of the four regions.

Plan: Put zero in the overlap. Since nothing can be in both events, all of \(P(A)\) belongs in A only and all of \(P(B)\) belongs in B only. Subtract the probabilities inside the circles from 1 to find neither.

Do: The overlap is zero, so A only is \(0.28\) and B only is \(0.37\). The probability of neither is:

P(\text{neither})=1-[0.28+0+0.37]=0.35

The completed diagram can be recorded as a region map:

A only	Both	B only	Neither
0.28	0	0.37	0.35

Check the regions by adding them: \(0.28+0+0.37+0.35=1.00\). The event probabilities also agree with the diagram: \(P(A)=0.28+0=0.28\), and \(P(B)=0.37+0=0.37\).

Conclude: In this model, 35% of residents use neither of these two commute options. The completed regions make the disjoint relationship explicit because the “both” region has probability zero.

Filling a Diagram for Overlapping Events

When events overlap, the intersection belongs to both event probabilities. To fill the diagram without counting that region twice, place \(P(A\cap B)\) in the overlap first. Subtract it from \(P(A)\) to get A only, and subtract it from \(P(B)\) to get B only. The union includes the three regions inside the circles; neither is the remaining probability outside them.

Formula: For overlapping or disjoint events, the individual circle totals include the intersection:

P(\text{A only})=P(A)-P(A\cap B),\qquad P(\text{B only})=P(B)-P(A\cap B)

The probability of neither is the amount left after all three inside regions are accounted for.

The overlap cannot be larger than either event. For example, \(P(A\cap B)\leq P(A)\), because every outcome in both events is also in \(A\). Similarly, \(P(A\cap B)\leq P(B)\). If subtracting the overlap produces a negative exclusive-region probability, the supplied values are inconsistent.

Worked Example: Filling an Overlapping-Events Diagram

Imagine an invented survey of app users. Let \(A\) mean that a randomly selected user enables notification reminders, and let \(B\) mean that the user enables a weekly summary. Suppose \(P(A)=0.52\), \(P(B)=0.41\), and \(P(A\cap B)=0.18\). Find all four region probabilities.

State: The events overlap because some users enable both features. We will place the given intersection in the shared region and calculate the other three regions.

Plan: Subtract the intersection from each event probability to find A only and B only. Add the three inside regions to find the union, then subtract that union probability from 1 to find neither.

Do: The exclusive regions are:

P(\text{A only})=0.52-0.18=0.34

P(\text{B only})=0.41-0.18=0.23

The union is the total of A only, both, and B only:

P(A\cup B)=0.34+0.18+0.23=0.75

Therefore, the probability of neither feature is \(1-0.75=0.25\). The completed region map is:

A only	Both	B only	Neither
0.34	0.18	0.23	0.25

Check the event totals: \(0.34+0.18=0.52\) for \(A\), and \(0.23+0.18=0.41\) for \(B\). The four regions sum to \(0.34+0.18+0.23+0.25=1.00\).

Conclude: In this invented survey model, 34% of users enable reminders but not the weekly summary, 23% enable the summary but not reminders, 18% enable both, and 25% enable neither.

Finding a Missing Overlap from the Union

Sometimes the given information includes \(P(A)\), \(P(B)\), and \(P(A\cup B)\), but not the intersection. In that case, use the general addition rule, which was introduced in The General Addition Rule. A Venn diagram offers a visual check: the union is the total of the three regions inside the circles, so the overlap must be the amount needed for that total to match the given union.

Formula: If the two event probabilities and their union are known, the intersection is:

P(A\cap B)=P(A)+P(B)-P(A\cup B)

After finding it, fill the two exclusive regions by subtracting the intersection from their event totals.

Worked Example: Recovering the Overlap from the Union

Suppose an invented outdoor club reports that \(P(A)=0.63\) for selecting a member who hikes, \(P(B)=0.48\) for selecting one who kayaks, and \(P(A\cup B)=0.82\) for selecting one who does at least one of these activities. Find the diagram regions.

State: We know each circle total and the total inside the circles, but not how much the circles overlap.

Plan: Use the general addition rule rearranged to find \(P(A\cap B)\). Then subtract the intersection from each circle total, and find neither as the complement of the union.

Do: The intersection is:

P(A\cap B)=0.63+0.48-0.82=0.29

Now subtract the overlap from each event probability:

P(\text{A only})=0.63-0.29=0.34,\qquad P(\text{B only})=0.48-0.29=0.19

The probability of neither activity is \(1-0.82=0.18\). Thus the region map is:

A only	Both	B only	Neither
0.34	0.29	0.19	0.18

Check the circle totals: \(0.34+0.29=0.63\) and \(0.19+0.29=0.48\). The inside regions sum to \(0.34+0.29+0.19=0.82\), and all four regions sum to 1.

Conclude: In this model, 29% of members do both activities, 34% hike but do not kayak, 19% kayak but do not hike, and 18% do neither.

Common Mistakes and AP Exam Tips

Putting the full event probability in the exclusive region. If events overlap, \(P(A)\) includes both A only and the intersection. Subtract the overlap before filling A only; do the same for \(B\).
Adding the circle totals without correcting for overlap. For overlapping events, \(P(A)+P(B)\) counts the shared region twice. Use the general addition rule or add the three inside regions once each.
Leaving out the outside region. A complete probability diagram accounts for the entire sample space. Find neither by subtracting the union probability from 1.
Treating separate-looking circles as proof of independence. Separate circles in a sketch represent disjoint events, not independent events. As discussed in Mutually Exclusive Versus Independent Events, disjoint events with positive probabilities are dependent.
Accepting negative or impossible region values. Each region must be between 0 and 1, and all four must sum to 1. Also check that the intersection is no larger than either event probability.
Giving numbers without identifying the regions. A clear answer labels A only, both, B only, and neither. For full credit, show how the overlap and remaining regions were calculated, then interpret the requested region in context.

A helpful routine is to place the overlap first, subtract it from each circle total, and finish with the outside region. Then verify both original event totals and the sum of all four regions. This makes the diagram a check on the arithmetic as well as a picture of the events.

Key takeaway: A two-event Venn diagram has four probability regions: A only, both, B only, and neither. For disjoint events, the overlap is zero. For overlapping events, place the intersection first, subtract it from each event total, and check that all four regions sum to 1.

Check Your Understanding

Sketch a two-event Venn diagram for each situation, label its regions, and show the probability calculations.

Events \(A\) and \(B\) are disjoint, with \(P(A)=0.22\) and \(P(B)=0.46\). Find the overlap and the probability of neither.
Suppose \(P(A)=0.70\), \(P(B)=0.55\), and \(P(A\cap B)=0.30\). Find A only, B only, and neither.
Given \(P(A)=0.45\), \(P(B)=0.38\), and \(P(A\cup B)=0.60\), find all four regions.
Suppose \(P(A)=0.25\), \(P(B)=0.40\), and \(P(A\cap B)=0.30\). Explain why these values cannot describe a valid two-event Venn diagram.
In your own words, explain why the overlap must be subtracted from \(P(A)\) to find A only when the events overlap.