A tree diagram shows every outcome of a sequence of events, with a probability on each branch. Multiplying along a path gives the probability of that exact sequence; adding the paths you want gives the probability of the event. Trees handle unequal probabilities and "without replacement" problems that sample space grids cannot.
The big picture
Two rules run everything. — this happens and then that happens — means multiply along the branches. — one path or another — means add the path probabilities.
The branches from any single point always add up to 1. When items are not replaced, the second set of branches changes: there is one fewer item in total, and one fewer of whatever was taken first. That makes the events .
What you'll learn
Draw one set of branches for each event. Write the outcome at the end of each branch and its probability on the branch. Check each set adds to 1.
For independent events — where one does not affect the other — .
The probability that Kai is late on any day is 0.2, independently of other days. Find the probability he is late on both Monday and Tuesday.
When more than one path gives the event you want, work out each path and add them.
"At least one" questions are often quickest with .
A biased coin lands heads with probability 0.6. It is flipped twice. Find the probability of exactly one head.
A fair coin is flipped 3 times. What is the probability of at least one head?
Tip — Say the path out loud: "red AND blue, OR blue AND red". AND means multiply, OR means add.
If an item is not replaced, the second-stage probabilities change. Reduce the total by one, and reduce the count of the colour already taken by one.
Fractions are usually clearer than decimals here — and do not simplify the branch fractions until the end.
A drawer has 4 black and 6 white socks. Two are taken at random without replacement. What is the probability that both are black?
Without replacement, taking a red sweet first makes the bag slightly "less red", so the chance of a second red falls from to . The second event depends on the first. With replacement, the bag is restored and the events are independent.
Think like an examiner
Tree diagram rules
Watch out for these
Stretch yourself
A bag contains counters, of which 3 are red. Two counters are taken without replacement. The probability that both are red is . Find .
Hint — Write the probability as and set it equal to .
Questions students ask
Key takeaways
Exam-style questions
3 original questions · 10 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.
The probability that it rains on a Saturday is 0.4. The probability it rains on a Sunday is 0.3. The events are independent.
A box contains 6 milk and 4 dark chocolates. Sam takes two at random without replacement.
Priya either walks or cycles to school. The probability she cycles is 0.35. If she cycles, the probability she is late is 0.1. If she walks, the probability she is late is 0.2.
Unlimited practice
A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.
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