A tree diagram maps every possible outcome of combined events, with each branch labelled with its probability. At GCSE, you multiply along a branch to find the probability of that specific sequence of events, then add the results of branches that satisfy your condition.
What does a GCSE tree diagram look like?
A tree diagram has a root (the start), a set of first-stage branches, and then a set of second-stage branches growing from each first-stage outcome. Each branch carries its probability.
Two rules govern every tree diagram:
- Multiply along the branches — the probability of following a particular path equals the product of all probabilities on that path.
- Add the results — if there are multiple paths that satisfy your condition, add those products together.
These two rules follow directly from the AND rule (multiply) and the OR rule (add) for independent events.
How do you draw a two-stage tree diagram?
Worked example: A bag contains 3 red and 2 blue counters. A counter is drawn, replaced, and a second counter is drawn. Draw a tree diagram and find the probability of getting two different colours.
Step 1 — first stage branches:
P(Red) = 3/5, P(Blue) = 2/5. Draw two branches from the start.
Step 2 — second stage branches:
Because the counter is replaced, the probabilities remain the same for the second draw. From each first-stage outcome, draw two branches again: P(Red) = 3/5, P(Blue) = 2/5.
Step 3 — calculate path probabilities:
| Path | Calculation | Probability |
|---|---|---|
| Red, Red | 3/5 × 3/5 | 9/25 |
| Red, Blue | 3/5 × 2/5 | 6/25 |
| Blue, Red | 2/5 × 3/5 | 6/25 |
| Blue, Blue | 2/5 × 2/5 | 4/25 |
Check: 9/25 + 6/25 + 6/25 + 4/25 = 25/25 = 1 ✓
Step 4 — answer the question:
Two different colours: Red, Blue OR Blue, Red. P(different) = 6/25 + 6/25 = 12/25
How do you handle without-replacement problems?
When items are not replaced, the probabilities on the second stage change based on what happened in the first stage.
Worked example: The same bag, but this time the counter is not replaced.
From the first draw:
- If Red drawn (3 left: 2 Red, 2 Blue from 4 total): P(Red 2nd | Red 1st) = 2/4, P(Blue 2nd | Red 1st) = 2/4
- If Blue drawn (4 left: 3 Red, 1 Blue from 4 total): P(Red 2nd | Blue 1st) = 3/4, P(Blue 2nd | Blue 1st) = 1/4
| Path | Calculation | Probability |
|---|---|---|
| Red, Red | 3/5 × 2/4 | 6/20 = 3/10 |
| Red, Blue | 3/5 × 2/4 | 6/20 = 3/10 |
| Blue, Red | 2/5 × 3/4 | 6/20 = 3/10 |
| Blue, Blue | 2/5 × 1/4 | 2/20 = 1/10 |
Check: 3/10 + 3/10 + 3/10 + 1/10 = 10/10 = 1 ✓
P(different colours) = 3/10 + 3/10 = 6/10 = 3/5
How do you handle "at least one" questions?
"At least one" means one or more. The quickest method is to use the complement:
P(at least one Red) = 1 − P(no Reds) = 1 − P(Blue, Blue)
Using the without-replacement example: P(at least one Red) = 1 − 1/10 = 9/10
This is faster than finding P(Red, Blue) + P(Blue, Red) + P(Red, Red) separately, though both methods give the same answer.
How do you use tree diagrams for three-stage events?
Three-stage trees have three sets of branches. The structure is the same — multiply along each path, add paths that satisfy the condition.
Example: A fair coin is flipped three times. Find P(exactly two heads).
Three-stage tree with H (prob 1/2) and T (prob 1/2) at each stage.
Paths giving exactly two heads: HHT, HTH, THH.
Each path probability = 1/2 × 1/2 × 1/2 = 1/8.
P(exactly two heads) = 1/8 + 1/8 + 1/8 = 3/8
What are the most common exam errors?
- Forgetting to update probabilities in without-replacement questions. The denominator decreases by 1 after each draw, and the numerator decreases by 1 only for the colour that was drawn.
- Adding instead of multiplying along a branch. Along a single path, all events must occur together (AND), so you multiply.
- Not listing all relevant paths. For "at least one success," students often forget one of the paths. The complement method avoids this risk entirely.
- Probabilities not summing to 1. Check that all end-of-branch probabilities add to exactly 1. If they do not, you have made an error somewhere in the diagram.
Frequently asked questions
When should I use a tree diagram rather than a two-way table?
Use a tree diagram when events happen in sequence (first draw, then second draw) or when the probabilities depend on what happened before (conditional probability). Use a two-way table when you have two attributes measured simultaneously on the same item (e.g. hair colour and eye colour). Tree diagrams are better for tracking order; tables are better for summarising frequency data.
Do the probabilities on each pair of branches from the same node have to add to 1?
Yes. From any node, the branches represent all possible outcomes at that stage — they are mutually exclusive and exhaustive, so their probabilities must sum to exactly 1. If you have three possible outcomes from a node (e.g. red, blue, green), all three branch probabilities must sum to 1. Use this as a built-in check after drawing each set of branches.
Can I leave probabilities as fractions throughout?
Yes, and for most tree diagram questions fractions are cleaner than decimals because they cancel neatly when multiplied. Write 2/4 as 1/2 only after the calculation is complete; keep all fractions unsimplified during the working so you can spot the common denominator when checking the total.
What is the difference between independent and dependent events on a tree diagram?
Two events are independent if the outcome of the first does not affect the probabilities of the second (e.g. tossing a coin twice). The branches from each node are identical. Two events are dependent if the first outcome changes the probabilities for the second (e.g. drawing without replacement). The branches from different first-stage nodes will carry different probabilities. Identifying this distinction is the first step in setting up any tree diagram correctly.
Want guided probability practice with Professor Pi? Visit AI Tutors for step-by-step help.