IB Math AI HL — Venn diagrams

Key Idea: Topic 4.6 builds on the probability rules from 4.5 by giving you two powerful visual tools: Venn diagrams (for showing how sets overlap and how probabilities relate) and tree diagrams (for sequential events where the outcome of the first affects the second). Both tools organise information — the key is knowing when to use each one.

✅ Venn diagrams

A ∩ B (intersection)

The overlap of the two circles — elements in both A and B. Fill this region first when given n(A∩B).

A ∪ B (union)

Everything in either circle (or both). n(A∪B) = n(A) + n(B) − n(A∩B).

A' (complement)

Everything outside circle A — neither in A nor in A∩B.

Only in A

n(A) − n(A∩B). The part of A not shared with B. Fill this after the intersection.

Outside both circles

n(S) − n(A∪B). Elements in neither set.

✅ Tree diagrams for conditional probability

With replacement

After the first draw, the item is put back. All probabilities on the second set of branches are the same as the first.

Without replacement

After the first draw, the item is removed. The denominator decreases by 1 on second branches, and the numerator changes depending on what was drawn first.

Reading the tree

Multiply along branches (AND). Add across outcomes that match (OR). Final probabilities in each path must sum to 1.

Example: Venn diagram: In a class of 30: 18 study French (F), 12 study Spanish (S), 7 study both. n(F∩S) = 7. n(F only) = 18−7 = 11. n(S only) = 12−7 = 5. n(neither) = 30−(11+7+5) = 7. P(F|S) = P(F∩S)/P(S) = (7/30)/(12/30) = 7/12 Tree diagram without replacement: 4 red, 3 blue balls. Draw two without replacement. P(red then blue) = (4/7) × (3/6) = 12/42 = 2/7

When filling a Venn diagram: always start with the intersection (A∩B), then fill in 'only A' and 'only B', then 'neither'. This order prevents double-counting. All probabilities in a tree diagram (summing across all final branches) must add to 1. Use this as a check.

Paper 1: You may be given a partially completed Venn diagram and asked to find a probability. Read n(A∩B) from the overlap region and use the conditional probability formula. Paper 2: Multi-stage probability problems almost always benefit from a tree diagram. Show all branches with their probabilities, then highlight the relevant paths.