Listing Outcomes
Listing Outcomes
Systematically listing all possible outcomes of an experiment ensures no outcome is missed or double-counted when calculating probabilities. The four main tools are: lists, tables, Venn diagrams and tree diagrams.
When to Use Each Tool
| Tool | Best for |
|---|---|
| Ordered list | Single-stage experiments with few outcomes |
| Two-way table / grid | Two-stage experiments with a small number of values each |
| Venn diagram | Two or three overlapping sets or categories |
| Tree diagram | Multi-stage experiments; especially with different probabilities at each stage |
Counting Principle
If one stage has \( m \) outcomes and another has \( n \) outcomes, the total number of combined outcomes is \( m\times n \). This extends to any number of stages.
Example: two dice each have 6 faces → \( 6\times6=36 \) total outcomes.
Tree Diagrams
Each branch represents one outcome of one stage. Multiply along branches to find the probability of a combined outcome; add across branches to find the probability of an event that can happen in several ways.
Worked Example — Tree Diagram
A bag has 3 red and 2 blue counters. One counter is drawn, replaced, then a second is drawn.
- \( P(RR)=\frac{3}{5}\times\frac{3}{5}=\frac{9}{25} \)
- \( P(RB)=\frac{3}{5}\times\frac{2}{5}=\frac{6}{25} \)
- \( P(BR)=\frac{2}{5}\times\frac{3}{5}=\frac{6}{25} \)
- \( P(BB)=\frac{2}{5}\times\frac{2}{5}=\frac{4}{25} \)
- Total: \( \frac{9+6+6+4}{25}=\frac{25}{25}=1 \) ✓
- \( P(\text{one of each colour})=P(RB)+P(BR)=\frac{12}{25} \)
Key Takeaways
- Systematic listing avoids missed or duplicate outcomes.
- Two stages with \( m \) and \( n \) outcomes → \( mn \) total combined outcomes.
- Tree diagram: multiply along branches (AND); add across branches (OR).
- All branch probabilities at each stage must sum to 1.
- Check: all combined outcome probabilities should sum to 1.