Venn Diagrams and Set Notation
Venn diagrams show which outcomes belong to different sets, including overlaps. Place each outcome in exactly one region, then use the relevant counts to calculate probabilities.
Foundation & Higher, with a Higher three-set reasoning extension. Use Probability Basics and Rules and Sample Spaces and Two-Way Tables first. Union and intersection notation is assessed by Edexcel at both tiers; OCR does not require the intersection, union or complement symbols at either tier. The meanings are explained here throughout.
Read the sets and their notation
The rectangle represents the universal set, written ℰ: everything under consideration. A circle represents a set, such as A. The overlap contains outcomes in both sets. Outcomes inside the rectangle but outside both circles still belong to the universal set.
| Notation | Meaning |
|---|---|
| x ∈ A | x is a member of A |
| x ∉ A | x is not a member of A |
| A ∩ B | Intersection: in both A and B |
| A ∪ B | Union: in A or B or both |
| A′ | Complement: in ℰ but not in A |
| ∅ | The empty set: no members |
| n(A) | Number of members of A |
The prime in A′ means “not A”; it is not an exponent. A set can be listed as A = {2, 4, 6}, or described by a rule, such as the even whole numbers from 1 to 6.
Place individual outcomes once
Let ℰ = {1, 2, …, 12}, A = even numbers and B = multiples of 3. Start with the overlap: 6 and 12 belong to both sets. Then fill the exclusive regions and finally the outside region.
Worked example — read probabilities from the elements
Choose an integer uniformly at random from 1 to 12. P(A ∩ B) = 2/12 = 1/6. A ∪ B contains eight numbers, so P(A ∪ B) = 8/12 = 2/3.
A′ contains 1, 3, 5, 7, 9 and 11, giving P(A′) = 1/2. The “B only” region has 3 and 9, so its probability is 1/6.
The numbers in this diagram are the outcomes themselves. Do not add their values: count how many outcomes are in the required region.
Use counts and fill the overlap first
Of 30 students, 18 study art, 15 study music and 7 study both. These set totals include the overlap. Let A mean art and B mean music.
Worked example — build and check the four regions
Put 7 in the overlap. Art only = 18 − 7 = 11. Music only = 15 − 7 = 8. At least one subject = 11 + 7 + 8 = 26. Neither = 30 − 26 = 4.
Check: 11 + 7 + 8 + 4 = 30. Selecting one student uniformly at random, P(exactly one subject) = (11 + 8)/30 = 19/30, whereas P(at least one) = 26/30 = 13/15.
These numbers are counts, unlike the individual elements in the previous diagram. Read the question to decide what the labels represent.
OR includes the overlap once
Adding P(A) and P(B) includes the overlap twice. Subtract it once to leave each outcome counted once: P(A or B) = P(A) + P(B) − P(A and B).
Worked example — apply the overlap rule
For the 30 students, P(art or music) = 18/30 + 15/30 − 7/30 = 26/30 = 13/15. This agrees with counting the three inner regions.
If two sets do not overlap, P(A and B) = 0 and the rule reduces to simple addition for mutually exclusive events.
“Neither” means the complement of the union: (A ∪ B)′. “Not both” means the complement of the intersection: (A ∩ B)′. They are different: here neither has count 4, but not both has count 30 − 7 = 23.
Higher extension — three overlapping sets
A third set creates more regions, but the principle stays the same: start with the deepest overlap and work outwards. If a question says “A and B”, that includes people in all three sets unless it explicitly says “A and B only”.
Among 40 people, totals are A:18, B:16, C:14. Pairwise totals are A and B:8, A and C:6, B and C:5. All three:3.
Worked example — subtract the centre before the pair-only regions
All three = 3. A and B only = 8 − 3 = 5; A and C only = 6 − 3 = 3; B and C only = 5 − 3 = 2.
A only = 18 − 5 − 3 − 3 = 7. Similarly B only = 6 and C only = 6. The seven inside regions total 32, leaving 8 outside.
Exactly two sets has count 5 + 3 + 2 = 10, so its probability is 10/40 = 1/4. The centre is excluded because it belongs to three sets.
Think Like a Mathematician
In the art/music example, P(art) + P(music) is 33/30, which exceeds 1. The individual probabilities are valid; the problem is that their sum counts seven students twice. A result above 1 can reveal an incorrect method rather than incorrect data.
Common Mistakes
Putting set totals in the “only” regions: subtract the overlap first.
Confusing OR with exactly one: ordinary mathematical OR includes both.
Ignoring the outside region: neither is still part of the universal set.
Three-set double counting: pairwise totals usually include the central overlap.
Practice
1. Explain A ∩ B, A ∪ B and A′ in words.
A ∩ B: both A and B. A ∪ B: A or B or both. A′: in the universal set but not in A.
2. Let ℰ = {1,2,…,10}, A = even numbers and B = numbers greater than 6. List A ∩ B and B only.
A ∩ B = {8, 10}. B only = {7, 9}.
3. For those sets, list the numbers in neither A nor B and find the probability of neither when choosing uniformly from ℰ.
Neither = {1, 3, 5}, so the probability is 3/10.
4. Of 36 people, 20 play tennis, 18 swim and 8 do both. Find tennis only, swimming only and neither.
Tennis only = 20 − 8 = 12; swimming only = 18 − 8 = 10. Union = 12 + 8 + 10 = 30, so neither = 6.
5. From those 36 people, find P(tennis or swimming) and P(exactly one activity).
OR = 30/36 = 5/6. Exactly one = (12 + 10)/36 = 11/18.
6. P(A) = 0.55, P(B) = 0.40 and P(A and B) = 0.15. Find P(A or B).
0.55 + 0.40 − 0.15 = 0.80.
7. Using the same probabilities, find P(neither) and P(not both).
Neither = 1 − 0.80 = 0.20. Not both = 1 − 0.15 = 0.85.
8. If A and B are mutually exclusive and P(A) = 0.3, P(B) = 0.5, what is P(A or B)? Must A and B be exhaustive?
The overlap is zero, so P(A or B) = 0.8. They are not exhaustive: probability 0.2 remains outside.
9. Higher: among 50 people, n(A)=23, n(B)=21, n(C)=19; n(A ∩ B)=10, n(A ∩ C)=9 and n(B ∩ C)=8; all three is 6. Find the probability of exactly two sets and the number in none.
Pair-only counts are 10−6=4, 9−6=3 and 8−6=2. Exactly two = 9/50. The single-only counts are 10, 9 and 8; all inside regions total 10+9+8+4+3+2+6=42. So P(exactly two)=9/50 and 8 are in none.
Frequently Asked Questions
Can a Venn diagram show probabilities instead of counts?
Yes. Each region may carry a probability; all disjoint regions inside the universal-set rectangle must then total 1. Check which kind of label the question uses.
Is set notation Higher-only?
No. Edexcel includes union and intersection notation in its Foundation teaching guidance. OCR does not require those symbols at either tier. Learn the meanings alongside the symbols so you can interpret both worded and symbolic questions.