Venn Diagrams
A Venn diagram sorts a group into overlapping sets, so 'A or B', 'A and B' and 'neither' can be read straight from it. This NSW Year 9 Advanced maths subtopic uses Venn diagrams and set notation to find probabilities (NESA outcome MA5‑PRO‑P‑01).
You will complete two-set and three-set Venn diagrams from survey data, find how many are in neither set, and calculate probabilities with \(A \cap B\), \(A \cup B\) and the complement \(A'\). You will find conditional probabilities when the choice is restricted, such as choosing only from basketball players, and use \(P(A \cup B)\) to find \(P(A \cap B)\). It is part of Year 9 Advanced Probability.
Every question with a fully worked solution.
- Venn Diagrams - Video - Probability Set Theory Venn Diagram Watch
Theory
A Venn diagram sorts every outcome into regions: in \(A\) only, in both, in \(B\) only, or in neither. Counting regions gives probabilities for "and", "or", "not" and "only". This Year 9 Advanced maths page also covers set notation, three sets and probabilities with a condition, on the path towards Advanced maths.
The rectangle is the universal set \(\xi\) (or \(S\)): everything being counted. Each circle is a set, a collection of distinct objects or outcomes.
\(A \cap B\), the intersection, is "\(A\) and \(B\)": the overlap. \(A \cup B\), the union, is "\(A\) or \(B\)" (or both): everything inside either circle. \(A'\) is the complement of \(A\): everything outside circle \(A\). \(n(A)\) is the number of elements in \(A\).
"Or" on its own is the inclusive or: \(A\), \(B\) or both. "\(A\) or \(B\) but not both" is the exclusive or. "Only \(A\)" is \(A \cap B'\). "At least one" means the union; "neither" is \((A \cup B)'\).
If \(A\) and \(B\) cannot both happen, they are mutually exclusive: the circles do not overlap and \(P(A \cap B) = 0\).
With a condition ("given that \(B\) happens"), only the outcomes in \(B\) count:
How to build a Venn diagram
- Start with the overlap. If it is unknown, find it: (circle totals added) \(-\) (number inside the circles).
- Subtract to get the "only" regions: the totals for \(A\) and \(B\) are whole circles and include the overlap.
- Find the outside: total \(-\) everything inside the circles.
- Three sets: fill the centre first, then the three "exactly two" regions, then the "only" regions, then the outside.
How to read a probability from it
- Translate the words into regions: and, or, only, neither, exactly one, at least two, not.
- Add those regions and divide by the total.
- With a condition ("of those who...", "given that..."), divide by the number in the condition, not the whole total.
- Unknowns: write each region with a pronumeral, make the regions add to the total, and solve.
(i) how many do both;
(ii) \(P(\text{exactly one})\);
(iii) \(P(D' \cap B)\).
(i) \(36 - 7 = 29\) students are inside the circles. Adding the circles counts the overlap twice.
| \(\text{both}\) | \(=\) | \(20 + 15 - 29 = \mathbf{6}\) |
Fill in: drama only \(20 - 6 = 14\), band only \(15 - 6 = 9\).
| \(\text{(ii) } P(\text{exactly one})\) | \(=\) | \(\dfrac{14 + 9}{36} = \mathbf{\dfrac{23}{36}}\) |
| \(\text{(iii) } P(D' \cap B)\) | \(=\) | \(\dfrac{9}{36} = \mathbf{\dfrac{1}{4}}\) |
\(D' \cap B\) is "not drama and band": the band-only region.
(i) List \(A \cap B\).
(ii) Find \(P(A \cup B)\).
(iii) Find \(P(A \cap B')\).
\(A = \{1, 2, 3, 4, 6, 8, 12, 24\}\) has \(8\) elements; \(B\) has \(12\) odd numbers.
(i) In both: \(A \cap B = \mathbf{\{1, 3\}}\).
| \(n(A \cup B)\) | \(=\) | \(8 + 12 - 2 = 18\) |
| \(P(A \cup B)\) | \(=\) | \(\dfrac{18}{24} = \mathbf{\dfrac{3}{4}}\) |
(iii) \(A \cap B'\) is the even factors: \(\{2, 4, 6, 8, 12, 24\}\).
| \(P(A \cap B')\) | \(=\) | \(\dfrac{6}{24} = \mathbf{\dfrac{1}{4}}\) |
(i) the number who use none;
(ii) \(P(\text{exactly two})\).
Start in the centre (\(4\)), then the "two only" regions, then the "one only" regions.
| \(N, S \text{ only}\) | \(=\) | \(10 - 4 = 6\) |
| \(N, D \text{ only}\) | \(=\) | \(7 - 4 = 3\) |
| \(S, D \text{ only}\) | \(=\) | \(9 - 4 = 5\) |
| \(N \text{ only}\) | \(=\) | \(38 - 6 - 3 - 4 = 25\) |
| \(S \text{ only}\) | \(=\) | \(30 - 6 - 5 - 4 = 15\) |
| \(D \text{ only}\) | \(=\) | \(25 - 3 - 5 - 4 = 13\) |
| \(\text{(i) none}\) | \(=\) | \(80 - 71 = \mathbf{9}\) |
| \(\text{(ii) } P(\text{exactly two})\) | \(=\) | \(\dfrac{6 + 3 + 5}{80} = \mathbf{\dfrac{7}{40}}\) |
(i) Find \(P(A \cap B)\).
(ii) Are \(A\) and \(B\) mutually exclusive?
(iii) Given that \(B\) happens, find the probability that \(A\) happens too.
| \(P(A \cup B)\) | \(=\) | \(1 - 0.25 = 0.75\) |
| \(P(A \cap B)\) | \(=\) | \(0.55 + 0.3 - 0.75 = \mathbf{0.1}\) |
(ii) No: \(P(A \cap B) = 0.1 \ne 0\), so they can happen together.
(iii) The condition "\(B\) happens" shrinks the sample space to circle \(B\). Of its \(0.3\), the part \(0.1\) is also in \(A\).
| \(P\) | \(=\) | \(\dfrac{0.1}{0.3} = \mathbf{\dfrac{1}{3}}\) |
Common pitfalls
Frequently asked questions
What does the upside-down U mean in a Venn diagram?
\(A \cap B\) is the intersection: the elements in both \(A\) and \(B\), which is the overlap of the circles.
What is the difference between union and intersection?
The union \(A \cup B\) is everything in \(A\) or \(B\) or both. The intersection \(A \cap B\) is only the elements in both.
What does A prime mean in probability?
\(A'\) is the complement of \(A\): every outcome that is not in \(A\). Its probability is \(1 - P(A)\).
How do you find the number in both sets of a Venn diagram?
Add the totals of the two sets, then subtract the number who are in at least one set. The extra amount is the overlap, which was counted twice.
How do you use a Venn diagram for conditional probability?
Restrict the sample space to the region in the condition. Count how many of those are also in the event you want, and divide by the number in the condition.
What are mutually exclusive events in a Venn diagram?
Mutually exclusive events cannot happen together, so their circles do not overlap and \(P(A \cap B) = 0\).