Put the Overlap in the Middle
In Using Two-Way Tables to Find Probabilities and Marginal Probabilities from a Table, you learned to distinguish probabilities for one event from probabilities for two events occurring together. A Venn diagram organizes those probabilities visually. For two events, the key is to place the probability of both events in the overlapping part of the circles before filling in the parts that belong to just one event.
Suppose \(A\) and \(B\) are events in a chance process. Draw a rectangle to represent the entire sample space, then draw two overlapping circles inside it. Label one circle \(A\) and the other \(B\). The overlap contains outcomes in both events. The part of circle \(A\) outside the overlap is “A only,” and the part of circle \(B\) outside the overlap is “B only.”
The rectangle matters because it represents all possible outcomes, not just the ones in the circles. Outcomes outside both circles belong to neither event. This tutorial focuses on placing the probabilities for \(A\), \(B\), and their overlap; the probability of “neither” is the subject of the next tutorial.
Fill the Shared Region First
If the probability of \(A\) is given, that probability includes both the part in \(A\) alone and the overlap. So the “A only” region is what remains after removing the overlap from \(P(A)\). The same reasoning applies to \(B\).
The union formula subtracts the overlap once because adding \(P(A)\) and \(P(B)\) counts the shared outcomes twice. It is also a useful check: the three values placed inside the circles should add to \(P(A\cup B)\).
A region map can help you organize the labels before drawing. Place each value in the matching part of the circles: the shared value in the center, and the exclusive values in the non-overlapping parts.
| Region | Where it goes | Probability |
|---|---|---|
| A only | Inside circle A, outside circle B | \(P(A)-P(A\cap B)\) |
| Both A and B | In the overlap | \(P(A\cap B)\) |
| B only | Inside circle B, outside circle A | \(P(B)-P(A\cap B)\) |
| Neither event | Inside the rectangle, outside both circles | Considered in the next tutorial |
When the probabilities are given as percentages, keep them in the same form throughout your calculations. For example, subtract 22% from 52% to get 30%; do not mix a percentage with a decimal unless you convert one first.
Worked Example: Start with the Event Probabilities
Worked Example: Start with the Event Probabilities
A library visitor is selected at random. Let \(A\) mean that the visitor borrowed an audiobook and \(B\) mean that the visitor borrowed an e-book. Suppose \(P(A)=0.52\), \(P(B)=0.40\), and \(P(A\cap B)=0.22\). Find the probability in each of the regions A only, both, and B only. Then find \(P(A\cup B)\).
Place the overlap: The probability of borrowing both types is already given, so put \(0.22\) in the intersection of the circles.
Find A only: The probability \(P(A)=0.52\) includes visitors in both the A-only region and the overlap. Subtract the shared probability:
Find B only: The probability \(P(B)=0.40\) also includes the overlap:
Find the union: Add the three probabilities inside the circles:
Conclude: Label the A-only region \(0.30\), the overlap \(0.22\), and the B-only region \(0.18\). The probability that a randomly selected visitor borrowed at least one of these two types is \(0.70\), or 70%.
Check the event totals against the diagram. The two regions inside \(A\) add to \(0.30+0.22=0.52\), and the two regions inside \(B\) add to \(0.18+0.22=0.40\). Both match the given probabilities.
Translate Counts into Venn Regions
Sometimes the information starts as counts rather than probabilities. As in the earlier tutorials on two-way tables, a count from the full group can be divided by the group total to express a probability. Once the event probabilities and overlap are known, use the same region calculations.
Worked Example: Build Regions from Survey Counts
In an invented survey of 200 students, 64 said they used the school shuttle at least once during a month, 90 said they rode a bicycle at least once, and 30 did both. Let \(A\) be the event that a student used the shuttle and \(B\) the event that a student rode a bicycle. Find the probabilities for A only, both, and B only, and find the probability of the union.
Convert the counts: Each count is from the same group of 200 students. The shuttle probability is \(64/200\), the bicycle probability is \(90/200\), and the overlap probability is \(30/200\).
Find A only: Of the 64 students who used the shuttle, 30 also rode a bicycle. That leaves \(64-30=34\) shuttle users in the A-only region. As a probability:
Find B only: Of the 90 students who rode a bicycle, 30 also used the shuttle. That leaves \(90-30=60\) in the B-only region:
Find the union: The number of students in either circle is \(34+30+60=124\). Dividing by 200, or adding the three region probabilities, gives:
Conclude: The Venn diagram has \(0.17\) in A only, \(0.15\) in the overlap, and \(0.30\) in B only. Thus, 62% of the surveyed students used the shuttle, rode a bicycle, or did both during the month.
The count check agrees: the A regions contain \(34+30=64\) students, and the B regions contain \(60+30=90\). The union contains 124 students, which is no more than the total of 200.
Find the Overlap When the Union Is Given
A problem may provide \(P(A)\), \(P(B)\), and \(P(A\cup B)\) instead of giving the overlap directly. The same union relationship can be rearranged to find the overlap. Then subtract that overlap from each event probability to get the two exclusive regions.
Worked Example: Find the Shared Probability
For two events \(A\) and \(B\), suppose \(P(A)=0.46\), \(P(B)=0.39\), and \(P(A\cup B)=0.68\). Find the probability of both events, A only, and B only.
Find the overlap: The union formula says that the union equals the sum of the event probabilities minus the overlap. Rearrange it by subtracting the union from the sum:
Find A only: Subtract the overlap from \(P(A)\):
Find B only: Subtract the overlap from \(P(B)\):
Check: The regions inside the circles add to \(0.29+0.17+0.22=0.68\), the given union probability. The regions within \(A\) add to \(0.29+0.17=0.46\), and those within \(B\) add to \(0.22+0.17=0.39\).
Conclude: Label the overlap \(0.17\), A only \(0.29\), and B only \(0.22\). The values reproduce all three probabilities in the question.
Check That the Diagram Is Possible
A Venn diagram should agree with every probability provided, and each region must have a probability from 0 to 1. These quick checks can reveal a subtraction error or inconsistent information before you finish your answer.
- The overlap cannot be greater than either event: \(P(A\cap B)\leq P(A)\) and \(P(A\cap B)\leq P(B)\).
- The A-only and B-only probabilities cannot be negative. If subtracting the overlap gives a negative value, check the inputs and calculations.
- The three regions inside the circles must add to the probability of the union: \(P(A\text{ only})+P(A\cap B)+P(B\text{ only})=P(A\cup B)\).
- The two regions inside circle \(A\) must add to \(P(A)\); the two regions inside circle \(B\) must add to \(P(B)\).
These checks are connected: when the overlap is placed correctly and each exclusive region is calculated correctly, the event totals and union should all agree. If they do not, return to the given information and verify which probability refers to the overlap and which refers to an entire circle.
Common Mistakes and AP Exam Tips
- Putting \(P(A)\) entirely in the A-only region. The full probability of \(A\) includes the overlap. Subtract \(P(A\cap B)\) to find A only.
- Adding \(P(A)\) and \(P(B)\) without accounting for the overlap. That counts outcomes in both events twice. For the union, subtract the overlap once, or add the three distinct inside regions.
- Putting the overlap outside the circles. Outcomes in \(A\cap B\) belong to both events, so they go in the shared part of the circles.
- Subtracting the wrong quantity. To find A only, subtract the overlap from \(P(A)\); to find B only, subtract the same overlap from \(P(B)\).
- Mixing counts and probabilities. Either subtract counts from counts or probabilities from probabilities. Convert counts to probabilities using the total group size before combining them with probability values.
For a clear AP response, name the overlap, show the subtraction used to find each exclusive region, and state what the union means in context. Include a check that the regions within each circle add to the given event probability. A labelled diagram is strongest when every number is placed in its proper region, rather than listed without indicating what it represents.
Check Your Understanding
For each question, identify which Venn diagram region the probability or calculation describes.
- Suppose \(P(A)=0.58\), \(P(B)=0.36\), and \(P(A\cap B)=0.14\). Find A only and B only.
- Using the probabilities in question 1, find \(P(A\cup B)\) by adding the three regions inside the circles.
- In a group of 150 people, 45 belong to event \(A\), 70 belong to event \(B\), and 20 belong to both. How many are in A only, and what is \(P(A\text{ only})\)?
- If \(P(A)=0.51\), \(P(B)=0.42\), and \(P(A\cup B)=0.73\), find \(P(A\cap B)\).
- Explain why the overlap must be subtracted when calculating \(P(A\cup B)\) from \(P(A)\) and \(P(B)\).