Skill: Represent sample spaces using tree diagrams

A tree diagram shows the stages of a random experiment from left to right. Each branch shows one possible result, and each complete path from the start to the end represents one outcome.

Example: A fair coin is tossed twice. Find the sample space and the probability of getting exactly one head.

Step 1: Identify the stages.

There are two stages:

1. The first coin toss
2. The second coin toss

For each toss, the possible results are $H$ (heads) and $T$ (tails).

Step 2: Draw branches for the first toss.

From the starting point, draw one branch for $H$ and one branch for $T$.

Step 3: Continue each branch for the second toss.

Each first-toss result can be followed by either $H$ or $T$.

Start
├── H
│   ├── H  → HH
│   └── T  → HT
└── T
    ├── H  → TH
    └── T  → TT

Step 4: List the complete paths.

The sample space is

$$S=\{HH, HT, TH, TT\}.$$

Each two-letter result is one ordered outcome. For example, $HT$ means heads first and tails second, while $TH$ means tails first and heads second. They are different outcomes because the order is different.

The paths are exhaustive because they include every possible result. They are nonoverlapping because one toss sequence cannot be both $HT$ and $TH$.

Step 5: Use the tree to find an event.

Exactly one head occurs on the paths

$$HT \quad \text{and} \quad TH.$$

Because the coin is fair, each branch has probability $\frac12$. Therefore, each complete path has probability

$$\frac12 \times \frac12=\frac14.$$

There are two paths with exactly one head, so

$$P(\text{exactly one head}) =\frac14+\frac14 =\frac12.$$

To make a tree diagram, show every possible result at each stage, follow every branch to an endpoint, and use the complete paths as the outcomes in the sample space.