Draw and label a tree diagram with probabilities on every branch
Multiply along branches to calculate the probability of combined outcomes (AND)
Add branch probabilities for OR outcomes and verify all paths sum to 1
Real-World Applications
Medical tests: A test for a disease has P(positive | disease) = 0.95 and P(positive | no disease) = 0.02. Tree diagrams map all four outcomes to find the true probability of disease given a positive result.
Quality control: Two components are tested. A tree diagram calculates P(both pass), P(exactly one fails), P(both fail) to plan replacement stock.
Weather: P(rain Monday) = 0.3. Tree diagrams extend this to two or three days to plan events.
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Drawing Tree Diagrams — Branches, Probabilities and Paths
A tree diagram shows every possible sequence of outcomes as a branching path. The probability is written on each branch. The three golden rules:
Branches from the same point must add up to 1
Multiply along branches to find the probability of a path (AND)
Add path probabilities for combined results that can happen in different ways (OR)
P(A and B) = P(A) × P(B) — multiply along branches
P(A or B) = P(path 1) + P(path 2) — add separate paths
Tree Diagram — Bag: 3 Red (R), 2 Blue (B) — Draw ONE marble
🔑 Rule to remember: Branches from the same point always add up to 1. Check this after drawing each set of branches.
✓ Quick Check 1 — Drawing and Reading Tree Diagrams
Question 1 of 10
Calculating Probabilities from Tree Diagrams
Once the tree is drawn, use it to calculate any probability by identifying the relevant paths and either multiplying (AND) or adding (OR). This block focuses on reading the numbers from the tree systematically.
Tree Diagram — Two Penalty Kicks: P(score) = 0.7, P(miss) = 0.3
Example 3 — P(both score) and P(at least one goal)
P(both score)
Follow the SS path: \(0.7 \times 0.7 = \mathbf{0.49}\)
P(at least one rainy day) = \(1 - P(\text{DD}) = 1 - 0.36 = \mathbf{0.64}\)
✓ When asked for "at least one", use the complement: 1 − P(none). It is almost always faster than adding all the favourable paths.
✓ Quick Check 2 — Calculating Probabilities from Tree Diagrams
Question 1 of 10
Interactive — Tree Diagram Builder
This tool always treats Event 1 and Event 2 as independent — if you pick the same bag for both, it models drawing from two separate identical bags (or replacing the item after the first draw), not drawing twice from one bag without replacing it.
First Event
Custom 1: A labelA countB labelB count
Second Event
Custom 2: A labelA countB labelB count
Practice: Combined Probabilities
Work out each combined probability as a decimal. Type your answer, then click Check.
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Question
Answer
Result
1
A fair coin is flipped twice. What is P(head, then head)?
2
A fair coin is flipped twice. What is P(at least one tail)?
3
A spinner has P(red) = 0.4. Spun twice (with replacement). What is P(red, red)?
4
A bag has P(red) = 0.4 and P(blue) = 0.6, with replacement. What is P(red, blue)?
5
P(win a game) = 0.2, played twice. What is P(win, win)?
6
Two independent events: P(A) = 0.5, P(B) = 0.3. What is P(A and B)?
7
A bag has P(yellow) = 0.3, with replacement, drawn twice. What is P(yellow, yellow)?
8
A spinner has P(blue) = 0.7. Spun twice. What is P(blue, blue)?
9
A bag has P(red) = 0.4, so P(not red) = 0.6, with replacement. What is P(not red, not red)?
10
Two independent events: P(A) = 0.2, P(B) = 0.3. What is P(A and B)?
⚠ Things to Watch Out For
Multiply along, add across: Multiply probabilities ALONG a branch path (AND). Add probabilities of different complete paths that give the same outcome (OR). Mixing these up is the most common error.
Branches from the same point must sum to 1: At every branching point, the probabilities on that set of branches must add up to exactly 1 — use this to check your tree.
Incomplete trees: Make sure every branch is drawn and labelled with a probability — a missing branch means a missing outcome.
Not simplifying the final fraction: After multiplying along a path, check whether the resulting fraction can be simplified.