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Venn diagrams

Covers AS91585

What a Venn diagram is for

A Venn diagram shows how events overlap. It is the best representation when a question describes memberships that can combine freely — people who own a cat, a dog, both or neither.
Use a Venn diagram when you are given information about unions, intersections and complements. Use a two-way table when the data are counts cross-classified by two variables, and a tree when the situation happens in stages.

The regions of a two-set diagram

A two-set diagram has exactly four regions, and together they account for everyone:
- $A$ only
- both $A$ and $B$ (the intersection)
- $B$ only
- neither — inside the rectangle, outside both circles.
The four regions are mutually exclusive and exhaustive, so their probabilities sum to 1. That is your check every time.

Filling one in — always start in the middle

The order matters, and doing it wrongly is the main source of errors:
- 1. Put the intersection in first — the "both" value.
- 2. Subtract to get the "only" regions: $A$ only $= P(A) - P(A \text{ and } B)$ .
- 3. Fill "neither" last, as $1$ minus everything else.
The circle labelled $A$ contains everyone in $A$ , including those also in $B$ . The number written in the " $A$ only" region is not $P(A)$ — a distinction worth checking every time you read a diagram.

Three-set diagrams

A three-set diagram has eight regions: the triple overlap, three double-only overlaps, three singles-only, and "none".
Always work outwards from the centre:
- Fill the triple intersection first.
- Then each pair intersection, subtracting the triple from each.
- Then each single, subtracting the two relevant pair-only regions and the triple.
- Then none.
Getting this order wrong double-counts the centre, which is the classic three-set error.

Reading probabilities off the diagram

In words	Regions to add
$P(A)$	$A$ only $+$ both
$P(A \text{ or } B)$	$A$ only $+$ both $+$ $B$ only (everything in either circle)
$P(A \text{ and } B)$	the intersection alone
$P(\text{neither})$	outside both circles
$P(\text{exactly one})$	$A$ only $+$ $B$ only
$P(A \text{ only})$	$A$ only
$P(A \mid B)$	intersection $\div$ ( $B$ only $+$ intersection)

Note the last row: a conditional probability from a Venn diagram means dividing by everything inside the conditioning circle, not by 1.

Mutually exclusive events on a Venn diagram

Mutually exclusive events are drawn as circles that do not overlap — there is no intersection region because $P(A \text{ and } B) = 0$ .
Independence cannot be seen by looking. It is a numerical relationship between the regions, not a visual feature, so it must be checked by calculation.

Worked ExampleA three-set Venn diagram

A survey of 240 Year 12 students asked which of three subjects they study: Biology (B), Chemistry (C) and Physics (P).

118 study Biology
96 study Chemistry
84 study Physics
47 study Biology and Chemistry
38 study Biology and Physics
41 study Chemistry and Physics
22 study all three

(a) Draw the Venn diagram and find how many study none of the three. (b) Find the probability a randomly chosen student studies exactly one of the three subjects. (c) Find the probability a student studies Physics given that they study Chemistry. (d) Are "studies Biology" and "studies Physics" independent?

(a) Filling the diagram

Step 1 — Centre first. All three: 22.

Step 2 — Pair-only regions. Each given pair figure includes the 22 in the centre, so subtract it.

$B \text{ and } C \text{ only} = 47 - 22 = 25$

$B \text{ and } P \text{ only} = 38 - 22 = 16$

$C \text{ and } P \text{ only} = 41 - 22 = 19$

Step 3 — Single-only regions. Each subject total includes everything inside its circle, so subtract the two pair-only regions and the centre.

$B \text{ only} = 118 - 25 - 16 - 22 = 55$

$C \text{ only} = 96 - 25 - 19 - 22 = 30$

$P \text{ only} = 84 - 16 - 19 - 22 = 27$

Step 4 — None.

$\text{inside the circles} = 55 + 30 + 27 + 25 + 16 + 19 + 22 = 194$

$\text{none} = 240 - 194 = 46$

Check: $194 + 46 = 240$ ✓

(b) Exactly one subject

Step 1 — Identify the regions. "Exactly one" means the three single-only regions: 55, 30 and 27.

$55 + 30 + 27 = 112$

Step 2 — Convert to a probability.

$P(\text{exactly one}) = \frac{112}{240} = 0.466666...$

(c) Physics given Chemistry

Step 1 — Identify the conditioning group. "Given they study Chemistry" restricts us to the whole Chemistry circle, which contains 96 students.

Step 2 — Find how many of those also study Physics. Inside the Chemistry circle, the Physics students are the $C$ and $P$ only region plus the centre:

$19 + 22 = 41$

(This is the given "Chemistry and Physics = 41", which is a useful check.)

Step 3 — Divide.

$P(P \mid C) = \frac{41}{96} = 0.427083...$

Compare with $P(P) = \frac{84}{240} = 0.35$ : taking Chemistry makes a student more likely to take Physics than a randomly chosen student is.

(d) Independence of Biology and Physics

Step 1 — Assemble the probabilities.

$P(B) = \frac{118}{240} = 0.491666... \qquad P(P) = \frac{84}{240} = 0.35$

$P(B \text{ and } P) = \frac{38}{240} = 0.158333...$

Note that $P(B \text{ and } P)$ uses the given 38, which includes the 22 taking all three — everyone in both circles counts.

Step 2 — Apply the test.

$P(B) \times P(P) = 0.491666... \times 0.35 = 0.172083...$

Step 3 — Compare.

$0.158333... < 0.172083...$

Step 4 — Interpret in context.

$P(P \mid B) = \frac{38}{118} = 0.322 \qquad \text{against} \qquad P(P) = 0.35$

Among Biology students, 32% take Physics, compared with 35% of all students — a small difference.

A plausible mechanism: students choosing three sciences are constrained by the number of option lines available, and Biology and Physics appeal to somewhat different interests (life sciences against physical sciences), with Chemistry sitting between them and pairing readily with either. Note that Chemistry and Physics were positively associated in part (c) while Biology and Physics are mildly negative — consistent with Chemistry acting as the common partner subject.

A caution on the size of the effect. The departure from independence here is small — 0.158 against 0.172 — and with 240 students it is well within what chance variation in subject choice could produce. It would be overstating the evidence to describe Biology and Physics as genuinely incompatible; the honest statement is that these data are close to independent, with a slight negative tendency.