A Venn diagram turns a probability problem into a picture. A rectangle is the sample space U; each event is a circle; overlaps show outcomes shared between events. Once it’s filled in, you can read off ∩ (overlap), ∪ (everything in the circles), ′ (outside) and even conditional probabilities at a glance. The golden rule for filling one in: start in the centre with the intersection and work outwards.
📘 What you need to know
The rectangle is the sample space U; each circle is an event; overlaps show shared outcomes.
Numbers can be frequencies (add to the total) or probabilities (add to 1).
A ∩ B = the overlap; A ∪ B = everything inside either circle; A′ = everything outside A.
Mutually exclusive events show instantly — their circles don’t overlap.
Independence can’t be seen — you must test it with P(A ∩ B) = P(A)P(B).
Fill from the centre out: put the intersection in first, then subtract to get the rest.
Conditional P(A|B) = (total in the B circle that’s also in A) ÷ (total in the B circle).
If a region is unknown, label it with algebra and use “everything sums to the total / to 1”.
Reading the regions
Every Venn question is really about which region you’re being asked for. Learn the four core ones.
the four key regions
∩ = overlap only; ∪ = both circles; ′ = outside; A ∩ B′ = the part of A that’s not in B (“A only”).
Frequencies or probabilities? If the numbers are counts, they add to the total n(U). If they’re probabilities, they add to 1. Check which before you start.
Filling one in: centre out
The reliable method: get the overlap first, then fill each circle so the parts add up correctly.
🧭 Recipe — completing a Venn diagram
Put the intersection in first (A ∩ B) — the centre.
Fill the rest of each circle: “A only” = n(A) − overlap; “B only” = n(B) − overlap.
Fill the outside (A′ ∩ B′) — the “neither” region.
Check it sums to the total (or to 1 for probabilities).
🤔 Why subtract the overlap when filling each circle?
The number n(A) counts everyone in A — including those also in B. If you wrote n(A) in the “A only” part, you’d double-count the overlap. So the part of A outside B is n(A) − n(A ∩ B). That’s why the intersection goes in first — everything else is measured relative to it.
🧠 Memory aid — “unknown region? use algebra”
If you can’t fill a region directly, call it x. Then form an equation from a known fact: all regions add to the total (or to 1), or P(A ∩ B) = P(A)P(B) if told independent. Solve for x.
Worked examples
WE 1
Fill in a Venn diagram with algebra
40 people are asked about sugar (S) and milk (M) in coffee: 21 have sugar, 25 have milk, 7 have neither. Find the number who have both.
let both = x; fill centre outsugar only = 21 − x, milk only = 25 − x, neither = 7.everything sums to 40(21 − x) + x + (25 − x) + 7 = 4053 − x = 40x = 13 have bothuse the total to form an equation for the unknown overlap.
Sugar only 8, both 13, milk only 12, neither 7 — total 40 ✓
WE 2
Read off a region
Using the completed diagram, find P(has sugar but not milk).
“sugar but not milk” = S ∩ M′ (sugar only)= 8 people out of 40P(S ∩ M′) = 8/40 = 1/5remember to write it as a fraction of the total.
WE 3
Conditional from a Venn diagram
Given a person who has sugar, find the probability they have milk.
restrict to the S circle (the given)total with sugar = 8 + 13 = 21of those, how many have milk?the overlap = 13P(M|S) = 13/21denominator = whole S circle; numerator = overlap.
WE 4
Find a union
Using the same diagram, find P(has sugar or milk).
S ∪ M = everything in the circles= 8 + 13 + 12 = 33P(S ∪ M) = 33/40or 1 − 7/40 (the 7 outside) = 33/40.
WE 5
Test independence from a Venn diagram
For events with P(A) = 0.5, P(B) = 0.4, and overlap P(A ∩ B) = 0.2 (read from a probability Venn diagram), are A and B independent?
independence can’t be seen — test itP(A)P(B) = 0.5 × 0.4 = 0.2P(A ∩ B) = 0.2 ✓ matchesyes — independenta Venn diagram shows exclusivity instantly, but never independence.
💡 Top tips
Fill the intersection first, then subtract to get the “only” regions.
Check the numbers sum to the total (frequencies) or to 1 (probabilities).
Unknown region → call it x and form an equation from the total or independence.
ConditionalP(A|B) = (overlap) ÷ (whole B circle).
“Only”, “but not”, “neither” all point to specific regions — translate carefully.
Mutually exclusive shows instantly (no overlap); independence must be tested.
⚠ Common mistakes
Writing n(A) in the “A only” region. Subtract the overlap first.
Forgetting the “neither” region outside both circles.
Not writing answers as a fraction of the total. A region count alone isn’t a probability.
Using the whole total as the denominator for a conditional — use the given circle.
Assuming independence from the picture. You must test it numerically.
Numbers not summing to the total / to 1 — always check at the end.
Next up — Tree Diagrams. Venn diagrams are great for “and/or/not” at a single stage; tree diagrams handle sequences of events. You’ll multiply along branches for intersections, add across outcomes for unions, and handle “with/without replacement” — the natural tool for conditional, multi-stage problems.
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