IB Maths AA HL
Topic 4 — Statistics & Probability
Paper 1 & 2
~7 min read
Venn Diagrams
A Venn diagram turns set-based probability into a picture: each region is a count or a probability, and you can read off any union, intersection, or complement at a glance. Once you can fill one in cleanly — including the centre piece for three sets — most “and / or / given that” questions become arithmetic. Build from the centre outwards and the rest falls into place.
📘 What you need to know
- Layout: a rectangle U for the sample space, with one circle per event inside.
- Numbers in regions = frequency or probability — frequencies sum to the total, probabilities sum to 1.
- A ∩ B: the overlap region (between the two circles).
- A ∪ B: anywhere inside either circle (including the overlap).
- A′: anywhere outside circle A — including the “neither” region outside both circles.
- Mutually exclusive: circles drawn separately, no overlap region.
- Conditional P(A | B): numerator = overlap, denominator = whole of circle B.
- Three-set Venn: ALWAYS start with the centre A ∩ B ∩ C and work outwards.
Reading the regions of a Venn
Intersection
A ∩ B
just the overlap region
Union
A ∪ B
everything inside either circle
Complement
A′
anywhere outside circle A
Conditional
P(A | B) = overlapcircle B
restrict the universe to circle B
Cardinal rule: P(A) is the WHOLE of circle A — including the overlap with B. Reading just the “A-only” region gives the wrong answer.
Building a Venn from data
Whether you’re given counts, probabilities, or a mix, the procedure is identical: place the centre value first, then fill the surrounding regions by subtraction.
Two-set region equations
n(A only) = n(A) − n(A ∩ B) | n(neither) = total − n(A ∪ B)
Three-set centre-out approach
pairwise-only region = n(A ∩ B) − n(A ∩ B ∩ C)
If the centre is unknown, label it x and solve using the total. The answer pops out from one linear equation.
Two-set vs three-set Venn diagrams
| Feature | Two-set Venn | Three-set Venn |
|---|
| Distinct regions | 4 (incl. “neither”) | 8 (incl. “none”) |
| Start filling at | A ∩ B | A ∩ B ∩ C (centre) |
| Subtraction layers | 1 — overlap from each circle | 2 — centre from each pairwise; pairwise from each circle |
| Common pitfall | forgetting “neither” | not subtracting the centre out of pairwise totals |
🧭 Recipe — filling in a Venn diagram
- Sketch the rectangle and the circles; label them clearly.
- Place the centre first: the all-sets intersection (A ∩ B for two sets; A ∩ B ∩ C for three).
- Subtract outwards: pairwise-only regions = full pairwise minus centre.
- Fill single-set-only regions: each circle’s total minus all its inner overlaps.
- Compute “outside all circles”: total minus everything inside the circles, then sense-check the sum.
Worked examples
WE 1Two-set Venn from frequencies
60 students were surveyed about their hobbies. 28 read books regularly, 22 play sports, and 10 do both. (a) Find how many do neither. (b) Find the probability that a randomly chosen student reads books but does not play sports.
(a) Build the Venn — start at the overlap
n(B ∩ S) = 10
n(B only) = 28 − 10 = 18
n(S only) = 22 − 10 = 12
n(neither) = 60 − (18 + 12 + 10) = 20
(b) P(reads books only) = 18 / 60
= 3/10 = 0.3
(a) 20 do neither; (b) P = 3/10
“reads books but not sports” = the B-only region, NOT the whole of circle B
WE 2Build a Venn from probabilities
In a music class, the probability a student plays piano is 0.40, the probability they play violin is 0.25, and the probability they play neither is 0.50. Find the probability that a randomly chosen student plays both.
Step 1: Find P(at least one) from the complement
P(P ∪ V) = 1 − P(neither) = 1 − 0.50 = 0.50
Step 2: Apply union formula
P(P ∪ V) = P(P) + P(V) − P(P ∩ V)
0.50 = 0.40 + 0.25 − P(P ∩ V)
P(P ∩ V) = 0.65 − 0.50 = 0.15
P(plays both) = 0.15
“neither” + “at least one” = 1 — always your bridge between the two sides of a Venn
WE 3Conditional probability from a Venn
In a company of 80 employees, 35 know Spanish, 28 know French, and 12 know both languages. Given that a randomly selected employee knows Spanish, find the probability that they also know French.
Reduce to circle S as the new sample space
n(S) = 35 (denominator)
n(S ∩ F) = 12 (numerator — overlap)
Apply conditional
P(F | S) = 12 / 35
P(F | S) = 12/35 ≈ 0.343
conditional on a Venn = (overlap region) ÷ (the “given” circle’s total)
WE 4Test independence using a Venn
A Venn diagram for events A and B shows the four regions with these probabilities: A only = 0.35, both = 0.15, B only = 0.15, neither = 0.35. Determine whether A and B are independent.
Step 1: Read off P(A) and P(B)
P(A) = 0.35 + 0.15 = 0.50
P(B) = 0.15 + 0.15 = 0.30
Step 2: Compute P(A) · P(B)
0.50 × 0.30 = 0.15
Step 3: Compare with P(A ∩ B)
P(A ∩ B) = 0.15 (centre region)
0.15 = 0.15 ✓
A and B ARE independent
remember P(A) is the WHOLE circle — A only PLUS the overlap
WE 5Three-set Venn — streaming services
A survey of 100 people asked about three streaming services. Results: 55 use Netflix, 40 use Disney+, 35 use Spotify. 20 use Netflix and Disney+, 15 use Netflix and Spotify, 12 use Disney+ and Spotify. 8 use all three. Find (a) how many use none of the three, (b) the probability a randomly chosen person uses exactly one service.
Step 1: Centre — all three = 8
Step 2: Pairwise-only regions (subtract centre)
N ∩ D only = 20 − 8 = 12
N ∩ S only = 15 − 8 = 7
D ∩ S only = 12 − 8 = 4
Step 3: Single-set-only regions
N only = 55 − (12 + 7 + 8) = 28
D only = 40 − (12 + 4 + 8) = 16
S only = 35 − (7 + 4 + 8) = 16
Step 4: “None” region
Inside total = 28+16+16+12+7+4+8 = 91
n(none) = 100 − 91 = 9
Step 5: P(exactly one)
n(exactly one) = 28 + 16 + 16 = 60
P(exactly one) = 60/100 = 3/5 = 0.6
(a) 9 use none; (b) P(exactly one) = 0.6
three-set Venn: ALWAYS centre first, then pairwise, then singles, then “none”
WE 6Algebra on a Venn — find every region
50 customers were surveyed. P(uses product A) = 0.6, P(uses product B) = 0.5, and P(uses at least one) = 0.8. Find the number of customers in each of the four Venn regions: both, A only, B only, neither.
Step 1: Find P(A ∩ B) using union formula
P(A ∩ B) = P(A) + P(B) − P(A ∪ B)
= 0.6 + 0.5 − 0.8 = 0.3
Step 2: Subtract for “only” regions
P(A only) = 0.6 − 0.3 = 0.3
P(B only) = 0.5 − 0.3 = 0.2
P(neither) = 1 − 0.8 = 0.2
Step 3: Multiply by 50 for counts
n(both) = 0.3 × 50 = 15
n(A only) = 0.3 × 50 = 15
n(B only) = 0.2 × 50 = 10
n(neither) = 0.2 × 50 = 10
15 / 15 / 10 / 10 (sum = 50 ✓)
always check the four region counts add back up to the total
💡 Top tips
- Always start at the centre and work outwards — this is non-negotiable for three-set Venns.
- Sense-check the total: all regions must sum back to the original total (or to 1 for probabilities).
- If the centre is unknown, set it as x and solve a linear equation using the total.
- Conditional shortcut: numerator is the overlap region, denominator is the whole “given” circle.
- Don’t forget the “neither” region outside all the circles — it’s part of the sample space.
⚠ Common mistakes
- Counting the overlap inside both circles — pairwise totals already include the overlap; subtract it once.
- Reading P(A) as just the “A only” region — P(A) covers the whole circle, including the overlap.
- Forgetting the centre when computing pairwise-only regions in a three-set Venn — always subtract it first.
- Skipping the “neither” region — even if it isn’t asked for, it stops the totals balancing.
- Mixing up frequencies and probabilities in the same Venn — pick one and stick with it.
Next: Tree Diagrams — the sequential cousin of the Venn. Where Venns excel at static “and / or” questions, trees handle multi-stage events (drawing without replacement, two-step processes, conditional chains) far more cleanly.
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