The regions of a Venn diagram
A Venn diagram shows sets as overlapping circles inside a rectangle. The rectangle is the universal set \(U\); each circle is one event. Two overlapping circles split the space into four regions.
- Region
- One part of a Venn diagram. For two sets \(A\) and \(B\) the four regions are: \(A\) only, \(B\) only, the overlap \(A \cap B\), and outside both (which is \((A \cup B)'\)).
The overlap in the middle holds the elements in both sets, the intersection \(A \cap B\). The area outside both circles but still inside the rectangle holds everything in neither.
Placing the data
The golden rule: fill the overlap first. The "both" number goes in the middle, then you subtract it from each circle's total to find the "only" parts. Filling the outer parts of the circles straight from the totals is the fastest way to double count the overlap.
The totals include the overlap
When a question says "18 play football", that \(18\) usually includes the ones who also play something else. Subtract the overlap before writing the football-only region, or your numbers will not add up to the group size.
In a class of 30, 18 play football (F), 12 play basketball (B), and 5 play both. Draw the Venn diagram and find the number who play neither, then \(P(F \cup B)\).
Reading probabilities off it
Once every region holds a number, a probability is just the count in the regions you want divided by the total. The addition rule is visible in the picture: adding \(P(A)\) and \(P(B)\) counts the overlap twice, so you subtract it once.
Checking the worked example against the rule: \(\dfrac{18}{30} + \dfrac{12}{30} - \dfrac{5}{30} = \dfrac{25}{30} = \dfrac{5}{6}\), which matches.
Common questions
Most Venn questions ask for one of these regions. Match the wording to the region and read off the count.
| Question asks for | Region to read |
|---|---|
| Both A and B | The overlap \(A \cap B\) |
| A or B (at least one) | Everything inside the circles, \(A \cup B\) |
| A but not B | The \(A\) only region |
| Neither | Outside both circles |
| Not A | Everything outside circle \(A\) |
Where this is assessed
Placing counts in the correct regions and computing probabilities is Criterion A (Knowing and understanding). A neat, fully labelled diagram with every region filled is Criterion C (Communicating). Sorting survey or club data into a Venn diagram to answer a real question is Criterion D (Applying in real-life contexts).
Check yourself
Use the class-of-30 diagram (football only \(13\), both \(5\), basketball only \(7\), neither \(5\)) for these.
1. What is the probability a pupil chosen at random plays football only? +
The football-only region holds \(13\) pupils out of \(30\). So \(P(\text{football only}) = \dfrac{13}{30}\), which does not simplify. \(\dfrac{13}{30}\).
2. Find the probability a pupil plays neither sport. +
The region outside both circles holds \(5\) pupils, so \(P(\text{neither}) = \dfrac{5}{30} = \) \(\dfrac{1}{6}\).
3. Of 40 people, 22 like tea, 18 like coffee and 10 like both. How many like neither? +
Overlap \(= 10\). Tea only \(= 22 - 10 = 12\); coffee only \(= 18 - 10 = 8\). At least one \(= 12 + 10 + 8 = 30\). Neither \(= 40 - 30 = \) \(10\).
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