IB Maths AA SLTopic 4 — ProbabilityPaper 1 & 2~10 min read
Venn Diagrams
A Venn diagram is the easiest way to “see” how two or more events overlap. Two circles, one rectangle around them — and suddenly all the AND/OR/NOT logic from probability becomes visual. They’re a Paper 1 & 2 favourite.
📘 What you need to know
A Venn diagram uses circles inside a rectangle (the sample space U) to show events.
Overlap between circles = intersection (A ∩ B) = “AND”.
Both circles together = union (A ∪ B) = “OR”.
Outside the circle = complement (A‘) = “NOT”.
Numbers in the regions can show either frequencies (must add to total) or probabilities (must add to 1).
Always fill in the intersection (centre) first, then work outwards.
If two circles do not overlap, the events are mutually exclusive.
What a Venn diagram looks like
The basics:
A rectangle shows the whole sample space (U) — every possible outcome.
Each circle represents an event.
Where two circles overlap, both events happen.
Outside the circles but inside the rectangle = “neither event happened”.
Anatomy of a 2-event Venn diagram
A 2-event Venn diagram has 4 regions: A only, both, B only, and neither. If you can fill in all 4, you can answer any probability question about A and B.
The middle of the overlap is where both events happen. Either circle alone (or both) is the union. Anything outside circle A is the complement of A. Three regions, three rules.
How to fill in a Venn diagram
This is the bit that earns marks in exams. Always work in the same order:
The 4-step method
Draw two circles inside a rectangle. Label them clearly (A, B) and label the rectangle U.
Fill in the intersection (centre) first. The centre overlap is the trickiest, so do it before anything else.
Subtract to find “A only” and “B only”. A only = n(A) − n(A ∩ B), and similarly for B only.
Fill in the outside (neither) last. This is total − everything else inside the circles.
🤔 Why work from the centre outwards?
If you put a number in the “A only” region first, you can’t be sure it doesn’t include the overlap. But once you’ve filled the centre, the rest is just subtraction: A only = (everything in A) − (overlap). That’s why the centre is always step 1.
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If you don’t know the centre — use algebra
If the question doesn’t give you the intersection directly, label it x and write the other regions in terms of x. The “everything must add to the total” rule will let you solve for x. (See worked example 1!)
Finding probabilities from a Venn diagram
Once your Venn is filled in, finding any probability is just counting:
From a frequency Venn diagram
P(event) = count of region(s) in eventtotal count
Conditional probability from a Venn
For P(A|B) — “A given B”:
Conditional from a Venn (the visual way)
Restrict to the B circle. The total of all numbers in B is your denominator.
Count what’s also in A. The number in the overlap is your numerator.
Divide: P(A|B) = (overlap) ÷ (B total).
This is exactly the “shrunken sample space” idea from the conditional probability note — but visual. The B circle becomes your new total, and you ask what fraction of it lies inside A.
Spotting mutually exclusive and independent events
Mutually exclusive — easy to spot
If the two circles don’t overlap, the events are mutually exclusive. The intersection region simply doesn’t exist (or is 0).
Independent — needs a check
You can’t tell from a glance whether events are independent — you have to test it. Use either:
P(A ∩ B) = P(A) × P(B), or
P(A|B) = P(A).
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Stuck on a Venn? Use algebra
If a question gives you incomplete info, label the unknown regions with letters (e.g. x, y) and write equations using the total and any other facts. For independent events, also use P(A ∩ B) = P(A) × P(B) as an extra equation.
Worked examples
WE 1
Build a Venn diagram and find probabilities
40 people are asked if they have sugar (S) and/or milk (M) in their coffee. 21 have sugar, 25 have milk, and 7 have neither.
(a) Draw a Venn diagram. (b) Find the probability a person has sugar but not milk. (c) Given that the person has sugar, find the probability they also have milk.
Set up algebra: let x = both sugar AND milk. Then “sugar only” = 21 − x and “milk only” = 25 − x.part (a) — venn diagramTotal = 40 across all 4 regions:(21 − x) + x + (25 − x) + 7 = 4053 − x = 40 ⟹ x = 13So:Sugar only = 21 − 13 = 8, Both = 13, Milk only = 25 − 13 = 12, Neither = 7part (b) — sugar but not milkSugar only region = 8 out of 40:P(S ∩ M’) = 840 = 15P(S ∩ M’) = 15part (c) — milk given sugarRestrict to sugar circle: 8 + 13 = 21 people.Of those, 13 also have milk.P(M | S) = 1321P(M | S) = 1321always start with the centre (overlap) — work outwards from there!
WE 2
Find probabilities from a given Venn diagram
The Venn diagram shows the number of students in a class who play football (F) and basketball (B). Find (a) P(F), (b) P(F ∩ B), (c) P(F ∪ B), (d) P(F | B).
Total = 10 + 6 + 8 + 6 = 30part (a) — p(f)F circle:10 + 6 = 16P(F) = 1630 = 815P(F) = 815part (b) — p(f ∩ b)Just the overlap:6P(F ∩ B) = 630 = 15part (c) — p(f ∪ b)Both circles together:10 + 6 + 8 = 24P(F ∪ B) = 2430 = 45part (d) — p(f | b)Restrict to B circle:6 + 8 = 14Of those, 6 also play F.P(F | B) = 614 = 37conditional = restrict total to the “given” circle, then count overlap
WE 3
Use probabilities (not frequencies) in a Venn
For events A and B: P(A) = 0.6, P(B) = 0.5, P(A ∩ B) = 0.2. Find (a) P(A only), (b) P(neither), (c) P(A ∪ B).
Fill in the centre first, then work outwards. All regions must add to 1.Centre (intersection):0.2A only:P(A) − P(A ∩ B) = 0.6 − 0.2 = 0.4B only:P(B) − P(A ∩ B) = 0.5 − 0.2 = 0.3Neither:1 − (0.4 + 0.2 + 0.3) = 0.1part (a)P(A only) = 0.4part (b)P(neither) = 0.1part (c)P(A ∪ B) = 0.4 + 0.2 + 0.3 = 0.9P(A ∪ B) = 0.9probability Venns work the same way, but regions sum to 1 instead of n
WE 4
Algebra approach for unknown intersection
In a class of 35 students, 20 have a phone, 18 have a laptop, and 5 have neither. How many have both?
Don’t know the intersection — call it x and use the total.Let x = number with both.Phone only:20 − xLaptop only:18 − xTotal = 35:(20 − x) + x + (18 − x) + 5 = 3543 − x = 35 ⟹ x = 88 students have bothsetting up x for the unknown intersection is the classic Venn algebra move
WE 5
Test independence from a Venn diagram
From the Venn in WE 2 (football and basketball, total 30): 16 students play football, 14 play basketball, and 6 play both. Are F and B independent?
Test: P(F ∩ B) = P(F) × P(B)?P(F):1630 = 815P(B):1430 = 715P(F ∩ B):630 = 15Compare:P(F) × P(B) = 815 × 715 = 56225 ≈ 0.249P(F ∩ B) = 15 = 0.20.2 ≠ 0.249, so they’re not equal.F and B are NOT independentVenns make the multiplication test easy — just count three regions and compare
💡 Top tips
Always fill in the intersection (centre) first. Everything else follows by subtraction.
If the intersection isn’t given, call it x and write all regions in terms of x. The total then gives you an equation.
Frequencies must add to n. Probabilities must add to 1. Always check.
For conditional probability, restrict to the “given” circle and count what’s also in the other event.
Don’t forget the “neither” region outside the circles. It’s part of the sample space too.
Mutually exclusive = circles don’t overlap. Independent can’t be seen — you have to test it.
For three-event Venns, work from the centre (the triple overlap) outwards in the same way.
Use a Venn even if the question doesn’t ask for one — it makes spotting errors much easier.
⚠ Common mistakes
Putting “P(A) = 21″ inside the A-only region. P(A) = 21 means the WHOLE A circle = 21, including the overlap. The “A only” piece is 21 − overlap.
Forgetting the “neither” region. Even if it’s small, it counts toward the total.
Filling the outer regions before the centre. Once you’ve assigned numbers to “A only”, you can’t redo them — fill the centre first.
For probability Venns, regions not summing to 1. If they don’t, your numbers are wrong somewhere.
Confusing A ∩ B‘ with A ∩ B. The first is “A but NOT B” — only the A-only sliver. The second is the overlap.
Using P(A ∪ B) = P(A) + P(B) when the circles do overlap. You’d be double-counting the intersection.
Trying to spot independence visually. You can’t tell from a Venn diagram alone — always test with the formula.
Treating the rectangle as the union. The rectangle is the whole sample space. The union is just the two circles together (without the outside).
Venn diagrams shine when events overlap. The next note covers tree diagrams — perfect for sequential events where one thing happens, then another (especially “without replacement” problems).
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