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Represent sample spaces using tree diagrams

A tree diagram represents a finite sample space for a compound experiment by showing each possible outcome at each stage, with complete paths from the first branch to the final result representing individual outcomes. It supports systematic enumeration, reveals whether outcomes have been omitted or repeated, and connects the number of equally likely paths to probability calculations; more advanced cases involving dependent or conditional probabilities are not included here.

Detailed Explanation: Represent sample spaces using tree diagrams

A tree diagram lists the possible results of a multi-step experiment. Each branch shows one possible result at a stage, and each complete path from start to finish represents one outcome in the sample space.

Example: A fair coin is tossed twice. What is the sample space, and what is the probability of getting exactly one head?

  1. Draw branches for the first toss: Heads (HH) and Tails (TT).
  2. From each of those branches, draw the possibilities for the second toss: HH and TT.
  3. Read each complete path from left to right.
Start
├── H
│   ├── H  → HH
│   └── T  → HT
└── T
    ├── H  → TH
    └── T  → TT

The sample space is

{HH,HT,TH,TT}.\{HH, HT, TH, TT\}.

There are 44 equally likely complete paths. The outcomes with exactly one head are HTHT and THTH, so there are 22 successful outcomes.

Therefore,

P(exactly one head)=24=12.P(\text{exactly one head})=\frac{2}{4}=\frac{1}{2}.

To check your tree, make sure every possible result is included once and that every outcome is a complete path from the start to the end.

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    Probability Sample Space - Definition to Sample Space List

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