🌳🔢 Branching into Probability: Mastering Tree Diagrams for Dependent Events

Tree diagrams are a **powerful visual tool** that help O Level students tackle multi-stage probability problems. When events are *dependent*—meaning the outcome of one stage affects the next—the branches of a tree diagram neatly keep track of the changing probabilities. Mastery of this skill not only boosts exam scores but also sharpens logical thinking used in sciences, business studies, and everyday decision-making.

Core Concept: Dependent Events in Tree Diagrams

Probabilities **change** from one branch to the next because outcomes are removed or altered.
Each complete path is found by **multiplying** along the branches, e.g. $P(A \text{ then } B)=P(A)\times P(B|A)$ .
To find the probability of an event that can occur in several ways, **add** the probabilities of all the relevant paths.

Worked Example: Drawing Marbles Without Replacement

Problem: A bag contains

4

red and

2

blue marbles. Two marbles are drawn one after another without replacement. What is the probability that both marbles are red? Solution (step-by-step): 1. **First branch:**

P(\text{red}_1)=\dfrac{4}{6}=\dfrac{2}{3}

. 2. **Second branch (after a red is removed):**

P(\text{red}_2\,|\,\text{red}_1)=\dfrac{3}{5}

. 3. **Multiply along the path:** \[ P(\text{red}_1 \cap \text{red}_2)=\frac{2}{3}\times\frac{3}{5}=\frac{2}{5}. \] 4. **State the answer:** The probability of selecting two red marbles is

\boxed{\dfrac{2}{5}}

. These four logical steps mirror the four key moves you’d sketch on a tree diagram. Practise drawing the branches to see how the fractions adjust after each pick—this habit will minimise careless mistakes in the exam and speed up your working.