🌳 Branching Out: Mastering Tree Diagrams for Independent Events!

Tree diagrams are a clear visual tool that help you map out **independent events** step-by-step. They simplify probability questions often seen in O Level exams, allowing you to avoid careless errors and pick up method marks with confidence.

Main Concept: Why Tree Diagrams Work

In a tree diagram, each branch represents an outcome and its probability. Because the events are **independent**, the probability on every branch stays the same no matter what happened before. To find the probability of a route through the tree, you multiply along the branches; to find the probability of different routes leading to the same overall event, you add their results.

Always label every branch with the outcome and its probability (e.g. $0.4$ for rain, $0.6$ for no rain).
Check that probabilities coming out of each node add up to $1$ to avoid mistakes.

Worked Example: Tossing Two Fair Coins

Problem: A fair coin is tossed twice. Find the probability of (a) getting two Heads, (b) getting at least one Head.

Step 1: Draw a tree with two levels. Each branch has probability $0.5$ for $H$ and $0.5$ for $T$ .
Step 2:
$P(\text{HH}) = 0.5 \times 0.5 = 0.25$
Step 3: Routes for "at least one Head": HH, HT, TH. Their combined probability is
$0.25 + 0.25 + 0.25 = 0.75$

Tree diagrams turn multi-step questions into bite-sized calculations. **Practise** drawing them quickly and checking branch totals, and you’ll secure easy marks in your next O Level paper!