Combined Events & Tree Diagrams

Spec reference: Probability
Key idea: Calculate probabilities for combined events using sample space diagrams, Venn diagrams and tree diagrams.

---

Sample space diagrams

List all possible outcomes in a table or grid.

Example

Two fair dice are rolled. Find P(total = 7).

Count outcomes summing to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6 outcomes

Total outcomes = 36

$$P(\text{total}=7)=\frac{6}{36}=\boxed{{\displaystyle \frac{1}{6}}}$$

---

AND / OR rules

$$P(A\text{ and }B)=P(A)\times P(B)\text{(independent events)}$$$$P(A\text{ or }B)=P(A)+P(B)\text{(mutually exclusive events)}$$

---

Tree diagrams

• Each branch shows a probability
• Multiply along branches for AND
• Add probabilities for OR (at the end)
• All branches from a node must sum to 1

Example

Bag has 4 red and 6 blue balls. Two drawn with replacement.

$$P(\text{both red})=\frac{4}{10}\times \frac{4}{10}=\frac{16}{100}=\boxed{{\displaystyle 0.16}}$$

---

Without replacement

When not replacing, the second probability changes.

Example

Same bag, drawn without replacement.

$$P(\text{both red})=\frac{4}{10}\times \frac{3}{9}=\frac{12}{90}=\boxed{{\displaystyle \frac{2}{15}}}$$

---

Venn diagrams

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

---

Test Yourself

Question 1 of 5

Two fair coins. P(two heads)?