Probability Trees

What you will learn

• Draw branches for successive events.
• Multiply along branches.
• Add separate successful routes.
• Update probabilities without replacement.
• Use tree diagrams for conditional situations.
• Use complements and given-that information when they are quicker than adding several routes.
• Count repeated independent routes for exactly-one and exactly-two event questions.

Key facts before you start

Why this matters

Tree diagrams make two-stage and three-stage probability questions much easier to follow.

Prior knowledge

You should already be comfortable with:

• Fractions.
• Multiplication.
• Adding fractions.
• Probability scale.

Probability trees infographic

Clear explanation

Main idea

For one complete route through a tree, multiply along the branches. If there is more than one successful route, find each route probability first, then add those route probabilities.

Method

With replacement, the second-stage probabilities stay the same. Without replacement, the total goes down and the category counts may change, so the second branch fractions usually change.

For at least one or not both questions, a complement can be quicker: find the probability of the opposite event and subtract it from 1.

If a later part gives information such as the first counter was red, start from the branch that matches that information. Do not keep routes that the condition has ruled out.

For three-stage independent trees, the same rules continue: multiply along one complete route. When many routes count as success, the complement is often cleaner than listing every successful route.

Answer tip

Write the event next to the route you need. For exactly one red, the successful routes are red then blue and blue then red, so both route probabilities must be included.

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For probability trees, marks usually come from drawing or completing the branches correctly, labelling replacement or no-replacement probabilities, multiplying along each route, adding alternative successful routes and simplifying or rounding the final probability as requested.

Common mistakes

• Adding along a route: multiply along one route; add only after route probabilities have been found.
• Forgetting replacement: with replacement probabilities stay the same, but without replacement the second-stage fractions usually change.
• Counting one route for exactly one event: exactly one red means red then not red and not red then red.
• Dropping branch labels: labels make it clear which probability belongs to which route.

Extension challenge

A bag contains 5 red counters and 3 blue counters. Two counters are taken without replacement. Draw the tree and find the probability of two red counters.

Exam-board guidance

Probability Trees appears within shared GCSE probability content. The same core method applies across boards: branch labels first, route calculations second, final probability last, with replacement decisions checked before any arithmetic.

Next lesson

Next, continue with Independent and Dependent Events.