Probability Tree Diagrams: Step-by-Step for GCSE

Probability tree diagrams are one of the highest-yielding topics on Higher tier and a regular Foundation appearance. Get the method right and you can pick up 6–8 marks across the paper in a few minutes.

What a tree diagram does

It maps every possible sequence of outcomes. Each branch shows the probability of that outcome happening given everything that’s happened so far. Multiply along a path to get the probability of that exact sequence. Add the probabilities of paths you want.

The general method

Tip: Always write probabilities on the branches, not the nodes. Examiners are looking at branch labels — that’s where the marks live.

1. Draw branches for the first event. Label probabilities.
2. Draw a fresh set of branches off each first-event node for the second event. Label.
3. Repeat for as many events as the question has.
4. To find P(outcome A) — multiply along the relevant path.
5. To find P(any of several outcomes) — multiply along each, then add.

Worked example: with replacement

A bag has 6 red and 4 blue counters. You pick one, note it, replace it, then pick another. P(both red)?

Probabilities on each branch:

• P(red on pick 1) = 6/10 = 3/5
• P(red on pick 2 | replaced) = 6/10 = 3/5 (same — that’s what “replacement” means)

P(both red) = 3/5 × 3/5 = 9/25.

Worked example: without replacement

Same bag. Pick one, don’t replace, pick another. P(both red)?

After removing one red:

• 5 red and 4 blue remain → 9 total
• P(red on pick 2 | first was red) = 5/9

P(both red) = 6/10 × 5/9 = 30/90 = 1/3.

Variant	Pick 1	Pick 2 (after red)	P(both red)
With replacement	6/10	6/10	9/25
Without replacement	6/10	5/9	1/3

Remember: The first pick is the same in both cases. Replacement only affects later picks.

The “at least one” shortcut

A bag has 5 red and 5 blue counters. You pick 3 with replacement. What is P(at least one red)?

The slow way: add P(exactly 1 red) + P(exactly 2 red) + P(exactly 3 red). Three calculations.

The fast way:

• P(at least one red) = 1 − P(no reds)
• P(no reds) = P(all blue) = (5/10)³ = 1/8
• Answer = 1 − 1/8 = 7/8

Always try the complement first. The question “at least one of X” almost always has a simpler “none of X” complement.

Conditional probability questions

Higher tier loves the structure: “Given that the first card was red, what’s the probability the second is also red?” That given is doing all the work — it tells you which branch you’re already standing on.

Did you know? “Given that” in maths means “we are restricted to the universe where this has happened.” It’s a denominator change, not a numerator one. The probability is P(event ∩ given) ÷ P(given).

Common errors

1. Forgetting to subtract from the total when calculating “at least one.” This trips up everyone at some point.
2. Replacing on the second branch by accident. If the question says “doesn’t replace,” update the denominator.
3. Not adding all relevant paths. “P(one red, one blue, in any order)” means add P(red then blue) and P(blue then red).
4. Forgetting to simplify, especially at the end of a 3-event tree where denominators get big (60, 120, 240…).

Drill them properly

Tree diagrams stick with practice, not reading. Try the dedicated topics in our Statistics quiz — probability is the largest of the four subject areas in the GCSE Statistics app, with subtopics specifically on tree diagrams, conditional probability and “at least one” questions.