🎯 Mastering Combined Events: Mutually Exclusive & Independent!

Ever wondered how likely it is for two things to happen together? From drawing cards to tossing coins, **combined events** are everywhere in O-Level questions. Mastering them not only boosts your marks but also sharpens logical thinking for real-life decisions!

Mutually Exclusive vs Independent Events

**Mutually exclusive events** can’t happen at the same time, so

P(A \cap B)=0

. **Independent events** don’t influence each other, so

P(A \cap B)=P(A)\times P(B)

. When events can occur together, the addition law applies:

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

Knowing which rule to use saves precious exam time!

If events are mutually exclusive, $P(A \cup B)=P(A)+P(B)$ .
If events are independent, $P(A \cap B)=P(A)\times P(B)$ .
Check wording: “or” usually means $\cup$ , “and” means $\cap$ .
Work with fractions or decimals consistently to avoid slips.

Worked Example: Rolling Two Dice

Problem: Two fair six-sided dice are thrown. Find (a) the probability that both show 3, (b) the probability that at least one shows 3.
Step 1: Define events: $A$ = first die shows 3, $B$ = second die shows 3.
Step 2: The dice are independent, so $P(A \cap B)=P(A)\times P(B)=\frac{1}{6}\times\frac{1}{6}=\frac{1}{36}$ .
Step 3: “At least one 3” means $A \cup B$ . Use the addition law:
$P(A \cup B)=P(A)+P(B)-P(A \cap B)=\frac{1}{6}+\frac{1}{6}-\frac{1}{36}=\frac{11}{36}$
.
Conclusion: Practice mixing the two rules in different contexts to become exam-ready!