Tree Diagrams (Edexcel GCSE Maths): Revision Notes

Tree Diagrams

Tree diagrams are a powerful visual tool for solving probability problems involving multiple events. They help you organise information clearly and calculate probabilities systematically, making even complex problems much more manageable.

Four key tree diagram facts

Understanding these fundamental principles will help you tackle any tree diagram question with confidence.

These four rules form the foundation of all tree diagram work and apply to every probability problem involving sequential events:

Mastering these rules will give you the confidence to approach any tree diagram problem systematically and accurately.

Sampling with replacement

When items are replaced after being selected, the probabilities remain the same for each subsequent selection. This creates a straightforward tree diagram where branch probabilities don't change.

In replacement scenarios, the key characteristic is that each selection is independent of previous selections. The probability of drawing a red disc remains $\frac{5}{8}$ for both the first and second draws because the first disc is returned to the bag. This means the composition of the bag stays the same throughout the process.

To solve these problems, you multiply along each branch to get the end probabilities, then add the relevant ones together. For example, finding the probability that both discs are the same colour involves adding $P(\text{both red}) + P(\text{both green})$.

Sampling without replacement

When items are not replaced, the probabilities change after each selection. This creates conditional probabilities where later selections depend on earlier outcomes.

Without replacement, the probabilities for the second selection depend on what happened in the first selection. If a red disc is drawn first, there are now fewer red discs and fewer total discs remaining, which changes the probability ratios for the second draw.

The key insight is that second-stage probabilities are conditional on first-stage outcomes. After drawing a red disc first, the probability of drawing red again becomes $\frac{4}{7}$ (since there are 4 red discs left out of 7 total), while the probability of drawing green becomes $\frac{3}{7}$.

Extra details for the tree diagram method

These additional techniques will help you handle more complex probability problems effectively.

Breaking problems into sequences

Always break complex probability questions into a sequence of separate events. Even if the question seems to involve simultaneous events, you can usually split it into sequential steps that are easier to analyse with a tree diagram.

Using partial diagrams

You don't always need to draw a complete tree diagram. Sometimes you can solve problems by drawing just the branches you need for your specific calculation.

Understanding conditional probabilities

Watch out for situations where probabilities change based on previous outcomes. This commonly occurs when sampling without replacement, but can also happen in other scenarios where earlier events affect later possibilities.

Handling "at least" questions

Questions asking for "at least" something can often be solved more easily by using the complement. The probability of "at least one" equals 1 minus the probability of "none".

For example: $P(\text{at least one six}) = 1 - P(\text{no sixes})$

This approach is particularly useful when calculating the probability of getting "at least one" of something would require adding many different outcomes together.

Worked example with conditional probability

Let's examine a problem involving tiles with letters, which demonstrates conditional probability in action.

Remember!