Cambridge (CIE) IGCSE · International Maths Extended
Tree Diagrams & Multiple Events
The big idea: when one thing happens and then another thing happens, you multiply along the branches — and whenever there are several different ways to get the result you want, you add those possibilities together.
Summary — Everything In This Chapter
- Probability tree diagrams map out repeated experiments with two outcomes each, so you can see every possible path an event can take.
- Combined probabilities let you calculate "one thing then another" without even drawing a tree — using the golden rule:
AND → multiply, OR → add.
- Independent events don't affect each other (like two separate dice rolls). Dependent events — especially "without replacement" problems — do.
- Conditional probability is the probability of something happening something else has already happened. It's calculated out of a smaller, restricted set of outcomes.
- Combined conditional probabilities combine both ideas: successive events where the second one's probability changes because of what happened in the first — most commonly seen in "without replacement" bag/card problems.
1. Probability Tree Diagrams
How do I draw a tree diagram?
Think of a tree diagram as a map of every possible path a two-stage (or more) experiment could take. Each "experiment" — like flipping a coin, drawing a card, or a set of traffic lights — has two outcomes: something happening (A) or it happening (not A).
You read a tree diagram left to right, following the branches. The top-most path all the way across gives you "A B". Every other path down the tree gives you a different combination.
1st experiment 2nd experiment
P(B) ──── B → P(A and B)
P(A) ──── A ┤
┌ └ P(not B) ─ not B → P(A and not B)
START ────┤
└ P(not A) ── not A ┤
├ P(B) ──── B → P(not A and B)
└ P(not B) ─ not B → P(not A and not B)
Every pair of branches coming from the same point must add up to 1. That's because P(not A) = 1 − P(A) — something either happens or it doesn't, there's no third option.
How do I find probabilities from a tree diagram?
Once your tree is drawn and labelled, there are just two moves:
Rule 1 — Multiply along the branches
P(1st outcome 2nd outcome) = P(1st) × P(2nd)
Follow one path from left to right, multiplying every probability you pass.
Rule 2 — Add between separate cases
P(AA BB) = P(AA) + P(BB)
If there's more than one path that satisfies what the question is asking, add those path-probabilities together.
If you're asked for the probability of "at least one" outcome, it's almost always quicker to calculate 1 − P(none) rather than listing and adding every single case that counts as "at least one".
Worked Example — Traffic Lights
A worker drives through two sets of traffic lights. P(1st is green) = 5/7, and P(2nd is green) = 8/9.
Step 1 — Find the "not" probabilities.
P(1st red) = 1 − 5/7 = 2/7. P(2nd red) = 1 − 8/9 = 1/9.
Step 2 — Multiply along each path.
P(G,G) = 5/7 × 8/9 = 40/63 | P(G,R) = 5/7 × 1/9 = 5/63
P(R,G) = 2/7 × 8/9 = 16/63 | P(R,R) = 2/7 × 1/9 = 2/63
Step 3(a) — P(both red)? That's simply the path P(R,R) = 2/63.
Step 3(b) — P(at least one red)?
23/63
23/63