Skill: Represent sample spaces using tree diagrams

A tree diagram shows the possible results of a random experiment in stages.

• Each branch represents one possible result at that stage.
• Follow a branch from left to right to complete the experiment.
• Each complete path ending at the far right is one outcome in the sample space.
• Intermediate points, such as the result after only the first stage, are not complete outcomes.

Example

A bag contains $2$ red counters and $2$ blue counters. Two counters are drawn without replacement. The color of each counter is recorded.

Step 1: Show the first draw

The first counter can be:

• Red: probability $\frac{2}{4}=\frac12$
• Blue: probability $\frac{2}{4}=\frac12$

Step 2: Show the second draw

The probabilities for the second draw depend on the first draw because the first counter is not replaced.

• If the first counter is red, then $1$ red and $2$ blue counters remain:

  • Red: $\frac13$
  • Blue: $\frac23$

• If the first counter is blue, then $2$ red and $1$ blue counter remain:

  • Red: $\frac23$
  • Blue: $\frac13$

The tree can be represented as:

Start
├── Red, 1/2
│   ├── Red, 1/3   → RR
│   └── Blue, 2/3  → RB
└── Blue, 1/2
    ├── Red, 2/3   → BR
    └── Blue, 1/3  → BB

The sample space is the set of complete paths:

$$S=\{RR,\ RB,\ BR,\ BB\}.$$

The single letters at the first branches are not outcomes because they do not describe both draws.

To find the probability of a complete path, multiply the probabilities along that path:

$$P(RR)=\frac12\times\frac13=\frac16$$ $$P(RB)=\frac12\times\frac23=\frac13$$ $$P(BR)=\frac12\times\frac23=\frac13$$ $$P(BB)=\frac12\times\frac13=\frac16$$

For example, the probability of drawing one red and one blue is found by adding the two relevant paths:

$$P(\text{one red and one blue}) =P(RB)+P(BR) =\frac13+\frac13 =\frac23.$$

Thus, tree diagrams help you list every complete outcome exactly once and calculate probabilities by multiplying along paths and adding suitable paths.