🌳 Probability Tree
The most powerful tool for solving multi-stage probability problems
🎯 What Is a Probability Tree?
A probability tree is a diagram that describes an experiment with multiple stages.
| 🔵 Nodes Decision/outcome points | ➡️ Branches Outcomes + probabilities | 🏁 Leaves Final outcomes |
📐 Tree Structure
Rules for building a tree:
- Each stage of the experiment = a new level in the tree
- Each outcome = a new branch
- Write the probability on each branch
- Sum of probabilities from each node = 1
⭐ The Two Important Rules
Multiplication Rule
Along a path
Multiply the probabilities
"and" = multiply
Addition Rule
Between paths
Add the probabilities
"or" = add
💡 Remember: along = × | across = +
✏️ Example 1: Tossing a Coin Twice
H = Heads | T = Tails
Probability calculations:
| Path | Calculation (multiply along) | Result |
|---|---|---|
| H,H | \(0.5 \times 0.5\) | 0.25 |
| H,T | \(0.5 \times 0.5\) | 0.25 |
| T,H | \(0.5 \times 0.5\) | 0.25 |
| T,T | \(0.5 \times 0.5\) | 0.25 |
Check: \(0.25 + 0.25 + 0.25 + 0.25 = 1\) ✓
✏️ Example 2: Drawing Without Replacement
Question: A box contains 3 red balls and 2 blue balls. Draw 2 balls without replacement.
Note! The probabilities change:
| Path | Calculation | Result |
|---|---|---|
| R,R | \(\frac{3}{5} \times \frac{2}{4}\) | \(\frac{6}{20} = \frac{3}{10}\) |
| R,B | \(\frac{3}{5} \times \frac{2}{4}\) | \(\frac{6}{20} = \frac{3}{10}\) |
| B,R | \(\frac{2}{5} \times \frac{3}{4}\) | \(\frac{6}{20} = \frac{3}{10}\) |
| B,B | \(\frac{2}{5} \times \frac{1}{4}\) | \(\frac{2}{20} = \frac{1}{10}\) |
Check: \(\frac{6+6+6+2}{20} = \frac{20}{20} = 1\) ✓
⚠️ Critical difference:
With replacement: the probabilities do not change between stages
Without replacement: the probabilities change! (denominator decreases by 1)
🔍 Using the Tree to Answer Questions
From the previous example (3 red, 2 blue, without replacement):
Question a: What is the probability that both balls are red?
Answer: Single path (R,R)
\(P(\text{R,R}) = \frac{6}{20} = \frac{3}{10}\)
Question b: What is the probability that the balls are different colours?
Answer: Two paths (R,B) or (B,R) – add them!
\(P(\text{different}) = \frac{6}{20} + \frac{6}{20} = \frac{12}{20} = \frac{3}{5}\)
Question c: What is the probability that at least one ball is red?
Answer (method 1): Three paths (R,R), (R,B), (B,R)
\(P = \frac{6}{20} + \frac{6}{20} + \frac{6}{20} = \frac{18}{20} = \frac{9}{10}\)
Answer (method 2 – complement): \(1 - P(\text{B,B}) = 1 - \frac{2}{20} = \frac{18}{20} = \frac{9}{10}\)
🔄 With Replacement vs Without Replacement
| With replacement | Without replacement | |
|---|---|---|
| What happens? | Draw, replace, draw again | Draw and do not replace |
| Probabilities | Do not change | Change! |
| Denominator | Stays constant | Decreases by 1 each stage |
| Example (3 red, 2 blue) | \(P(\text{R,R}) = \frac{3}{5} \times \frac{3}{5} = \frac{9}{25}\) | \(P(\text{R,R}) = \frac{3}{5} \times \frac{2}{4} = \frac{6}{20}\) |
✅ Self-Check
Sum of all leaf probabilities = 1
If it doesn't equal 1, there is a calculation error!
Sum of probabilities from each node = 1
All branches from the same node must sum to 1
💡 Tips for the Exam
1️⃣ Always draw!
A tree helps you see all possibilities and prevents errors
2️⃣ Write probabilities
Write the probability on every branch
3️⃣ Identify with/without
Check if there is replacement – it affects the probabilities!
4️⃣ Mark paths
Highlight the paths relevant to the question
📝 Summary
| Along a path | → | Multiply × |
| Between paths | → | Add + |
Sum of all leaves = 1 | Sum of branches from a node = 1