Tree Diagrams - Visualising Multi-Step Probability

A tree diagram is a visual tool for listing all outcomes of a multi-step probability experiment and calculating probabilities by multiplying along branches. Tree diagrams make it easy to see every possibility at once and to apply both the multiplication rule and the addition rule systematically.

The branching structure of a probability tree is not just a teaching device – it is the same fundamental idea behind “game trees” in computer science, where every possible move and countermove in a game is mapped out as a branching diagram. The American mathematician and engineer Claude Shannon, better known as the father of information theory, wrote a landmark 1950 paper explaining how a computer could be programmed to play chess by searching through exactly this kind of tree of possible move sequences. Modern chess engines and game-playing AI systems still search enormous versions of the same branching tree structure you use here to work out P(bus is late) or P(both same colour).

How a Tree Diagram Works

Each branch represents one possible outcome at a given step.
The probability is written on the branch.
To find the probability of a path through the tree, multiply the probabilities along its branches.
To find the probability of an event made up of several paths, add the path probabilities.

The Two Rules

Along branches (AND): Multiply probabilities.
Between paths (OR): Add probabilities.

Worked Examples

A coin is tossed twice. Draw a tree diagram and find P(exactly one head).

Toss 1	Toss 2	Outcome	Probability
H (1/2)	H (1/2)	HH	1/2 × 1/2 = 1/4
H (1/2)	T (1/2)	HT	1/2 × 1/2 = 1/4
T (1/2)	H (1/2)	TH	1/2 × 1/2 = 1/4
T (1/2)	T (1/2)	TT	1/2 × 1/2 = 1/4

P(exactly one head) = P(HT) + P(TH) = 1/4 + 1/4 = 1/2. Check: all paths sum to 1. ✓

A bag has 3 red and 2 blue balls. Two balls are drawn WITHOUT replacement. Find P(both same colour).

1st draw	2nd draw	Outcome	Probability
R (3/5)	R (2/4)	RR	3/5 × 2/4 = 6/20
R (3/5)	B (2/4)	RB	3/5 × 2/4 = 6/20
B (2/5)	R (3/4)	BR	2/5 × 3/4 = 6/20
B (2/5)	B (1/4)	BB	2/5 × 1/4 = 2/20

P(both same) = P(RR) + P(BB) = 6/20 + 2/20 = 8/20 = 2/5. Check: 6+6+6+2 = 20. ✓

P(it rains) = 0.3. If it rains, P(bus is late) = 0.7. If it does not rain, P(bus is late) = 0.2. Find P(bus is late).

P(rain and late) = 0.3 × 0.7 = 0.21.
P(no rain and late) = 0.7 × 0.2 = 0.14.
P(bus is late) = 0.21 + 0.14 = 0.35.

Key Takeaways

Write probabilities on the branches, outcomes at the end of each path.
Multiply along branches (AND rule) to find each path's probability.
Add path probabilities (OR rule) when an event can happen via more than one path.
All end-of-path probabilities must sum to 1 – use this as a check.

Practice: Tree Diagrams

Two-Step Tree Diagrams

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