Use tree diagrams for sample spaces

A tree diagram represents the possible outcomes of a multistep chance experiment by showing each choice as a branch from the preceding choice; complete paths represent distinct outcomes in the sample space. The diagram supports systematic counting and identifying equally likely outcomes, including cases where choices differ at successive stages, while distinguishing complete outcomes from intermediate events; these ideas support calculating simple probabilities as favorable outcomes over total equally likely outcomes.

Detailed Explanation: Use tree diagrams for sample spaces

A tree diagram helps you list every possible outcome of a chance experiment with more than one step.

Each branch shows one choice or result.
Follow a branch all the way to the end to get one complete outcome.
Count complete paths, not just the branches in the middle.

Example: A coin is flipped and then a number cube is rolled. What is the probability of getting heads and an even number?

Step 1: Make the first branches

The coin can land on heads or tails:

Start
├── Heads
└── Tails

Step 2: Add the second branches

After each coin result, the number cube can show $1,2,3,4,5,$ or $6$ .

Start
├── Heads
│   ├── 1
│   ├── 2
│   ├── 3
│   ├── 4
│   ├── 5
│   └── 6
└── Tails
    ├── 1
    ├── 2
    ├── 3
    ├── 4
    ├── 5
    └── 6

Step 3: List the complete outcomes

The complete paths are

(H,1),(H,2),(H,3),(H,4),(H,5),(H,6)

and

(T,1),(T,2),(T,3),(T,4),(T,5),(T,6).

There are $12$ complete outcomes altogether. The word “Heads” by itself is not a complete outcome because the number cube result is still missing.

Step 4: Find the favorable outcomes

We need heads and an even number. The favorable paths are

(H,2),(H,4),(H,6).

There are $3$ favorable outcomes.

Since all $12$ complete outcomes are equally likely,

P(\text{heads and even})=\frac{\text{favorable outcomes}}{\text{total outcomes}} =\frac{3}{12} =\frac14.

So, the probability is $\boxed{\frac14}$ .

Learn by doing: Use tree diagrams for sample spaces

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Probability Sample Space - Sample Space Count and Favourable Outcomes Definition to Probability

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