Venn and Tree Diagrams (AI SL)
Venn diagrams and tree diagrams are the two pictures IB examiners lean on to turn a wordy probability scenario into something you can actually calculate. A Venn diagram sorts one fixed group into overlapping categories; a tree diagram tracks a sequence of draws or trials one after another. This page focuses on reading and building both, including the with/without-replacement distinction that trips up most students. It's part of the broader Probability topic.
30 questions on this sub-topic.
Combining events
Covered under IB syllabus reference SL4.6: use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities, including combined events and probabilities with and without replacement.
Multiplying along a branch
\(P(A\cap B) = P(A)\times P(B\mid A)\)
Multiply the probabilities you meet travelling along one path of a tree diagram. For independent draws (with replacement), \(P(B\mid A)\) is just \(P(B)\).
Union from a Venn diagram
\(P(A\cup B) = P(A) + P(B) - P(A\cap B)\)
Add the two regions, then subtract the overlap once so it isn't counted twice. In the formula booklet.
Need the full formula reference table and the mutually-exclusive-vs-independent comparison? See Probability.
Worked examples
A bag contains 4 red and 6 blue counters. Two counters are drawn, with replacement.
Find the probability that both are red.
Worked solution
There are \(4+6=10\) counters, so \(P(\text{red})=\dfrac{4}{10}=0.4.\) A1
The counter is returned, so the second draw has the same probabilities. For independent events multiply: M1
\(P(RR)=0.4\times0.4=0.16.\) A1
From a standard 52-card deck, two cards are drawn without replacement.
Find the probability both are hearts.
Worked solution
13 hearts in 52: \(P=\dfrac{13}{52}.\) A1
12 hearts left in 51: \(P=\dfrac{12}{51}.\) M1
\(P(\text{both hearts})=\dfrac{13}{52}\times\dfrac{12}{51}=\dfrac{156}{2652}.\) M1
Step 3b - simplify. \(\dfrac{156}{2652}=\dfrac{1}{17}\approx 0.0588.\) A1
Using the previous bus scenario, find the total \(P(\text{miss class})\).
Worked solution
\(0.2(0.7)+0.8(0.1)\) M1
\(=0.22.\) A1
Common mistakes
- Forgetting that "without replacement" changes later branches. If an item is removed and not put back, the totals and probabilities on the next branch of a tree diagram must be updated - they are not the same as the first draw.
- Adding along a branch instead of multiplying. Multiply probabilities along a single path through a tree diagram; only add the probabilities of separate paths that lead to the same outcome.
- Leaving Venn-diagram regions unlabelled before calculating. Fill in the intersection first, then work outward to each "only" region - trying to read probabilities off an incomplete diagram is where most arithmetic slips creep in.
Ready to practise properly?
30 Venn and tree diagram questions, marked instantly like the real exam.
Quick answers
When should I use a tree diagram instead of a Venn diagram?
Use a tree diagram for events that happen in sequence, like two draws from a bag one after the other. Use a Venn diagram when you're looking at how two categories overlap within one fixed group, like students who study French and/or Spanish.
Do I multiply or add probabilities on a tree diagram?
Multiply along a single branch to find the probability of that exact sequence of outcomes. Add the probabilities of separate branches only when they lead to the same final result you're interested in.