Probability tree diagrams are a brilliant visual tool for tackling questions involving combined events—where more than one thing happens, like flipping a coin twice or picking two sweets from a bag. If you're studying for your AQA, Edexcel, or OCR GCSE, you'll definitely need to know how to use them. This guide will walk you through the process, step-by-step.
What is a Tree Diagram and When Do You Use It?
A tree diagram is a way of mapping out all the possible outcomes of a sequence of events. Each branch represents a possible outcome, and we write the probability of that outcome along the branch.
You should use a tree diagram when:
Events happen one after another (e.g., first pick, then second pick).
You are dealing with combined events (e.g., the probability of A and B happening).
- The probabilities change after the first event (dependent events), like picking counters without replacement.
The Step-by-Step Walkthrough
Let's work through a classic GCSE-style problem.
> Example Problem: A bag contains 5 green counters and 3 red counters. Ria takes a counter at random, notes its colour, and does not put it back. She then takes a second counter at random. What is the probability that she takes two counters of the same colour?
Step 1: Identify the Events and Outcomes
First, break down what's happening.
Event 1: The first counter is taken. Outcomes: Green (G) or Red (R).
- Event 2: The second counter is taken. Outcomes: Green or Red.
Crucially, the second event depends on the first because the counter isn't replaced. The total number of counters changes.
Step 2: Draw the First Set of Branches
Draw a little 'tree' starting from a point on the left.
Draw two branches for the first event: one labelled G, one labelled R.
Write the probabilities on the branches.
P(First is Green) = 5 green out of 8 total = 58.
- P(First is Red) = 3 red out of 8 total = 38.
flowchart TD
Start["Start"] --> |5/8| G1["First: Green"]
Start --> |3/8| R1["First: Red"]
Step 3: Draw the Second Set of Branches
From the end of each first branch, draw the branches for the second event. The probabilities here are conditional.
If the first was Green: There are now 4 green and 3 red left (7 total).
P(Second is Green | First was Green) = 47.
- P(Second is Red | First was Green) = 37.
If the first was Red: There are now 5 green and 2 red left (7 total).
P(Second is Green | First was Red) = 57.
- P(Second is Red | First was Red) = 27.
flowchart TD
Start["Start"] --> |5/8| G1
Start --> |3/8| R1
G1 --> |4/7| G2["Outcome: G then G"]
G1 --> |3/7| R2["Outcome: G then R"]
R1 --> |5/7| G3["Outcome: R then G"]
R1 --> |2/7| R3["Outcome: R then R"]
Step 4: Calculate Probabilities of Specific Combined Events
To find the probability of a sequence of events, you multiply the probabilities along the branches.
P(G then G) = (58) × (47) = 2056.
P(G then R) = (58) × (37) = 1556.
P(R then G) = (38) × (57) = 1556.
- P(R then R) = (38) × (27) = 656.
Key Check: The sum of all these end probabilities should be 1. 20+15+15+656 = 5656 = 1. Perfect!
Step 5: Answer the Question
Our question asks for the probability of two counters of the same colour. This means either (G then G) or (R then R).
For 'OR' questions where outcomes are mutually exclusive, you add the probabilities.
P(Same colour) = P(G then G) + P(R then R) = (2056) + (656) = 2656.
Always simplify your final answer: 2656 = 1328.
Key Rules to Remember
Multiply along the branches to find 'AND' probabilities (this event and that event).
Add the probabilities of different branches to find 'OR' probabilities (this event or that event).
Probabilities on branches coming from the same point must add up to 1. This is a great way to check your work.
- With replacement vs. Without replacement: This is the biggest pitfall! If an item is replaced, the probabilities for the second event are the same as the first. If not, they change.
A Quick Example with Independent Events
For independent events (like coin flips), the second event probabilities don't change.
> Problem: A fair coin is flipped twice. Find the probability of getting a Head and a Tail (in any order).
flowchart TD
Start["Start"] --> |1/2| H1["First: Head"]
Start --> |1/2| T1["First: Tail"]
H1 --> |1/2| H2["Outcome: H then H"]
H1 --> |1/2| T2["Outcome: H then T"]
T1 --> |1/2| H3["Outcome: T then H"]
T1 --> |1/2| T3["Outcome: T then T"]
P(Head and Tail in any order) = P(H then T) + P(T then H) = (12 × 12) + (12 × 12) = 14 + 14 = 12.
Top Tips for GCSE Exam Success
Draw neatly! A messy tree is hard to follow and leads to mistakes.
Label everything clearly: Write the event and the probability on each branch.
Show your working: Write the multiplication sums next to the final outcomes. Even if your arithmetic slips, you'll get method marks.
Check your fractions add to 1 at the end of the branches—it's a quick verification.
- Remember to simplify your final answer where possible.
Practice Makes Perfect
The best way to get confident with tree diagrams is to practise. Look for questions involving:
Sports matches (win/draw/lose).
Quality control (faulty/not faulty).
Weather probabilities.
- Drawing cards or counters from a bag (with and without replacement).
Mastering tree diagrams will give you a huge boost in the probability section of your GCSE paper. Break the problem down, follow the steps, and you'll be able to tackle even the trickiest combined events question. Good luck!
- Want more practice? Search for past paper questions from your exam board (AQA, Edexcel, OCR) on 'probability tree diagrams' or 'conditional probability'.