Form 4 Mathematics Bab 9: Probability of Combined Events
A combined event is an event formed by combining two or more individual events, either sequentially or simultaneously.
The sample space, $S$, of combined events can be represented using:
Two events $A$ and $B$ are independent if the occurrence of event $A$ does not affect the probability of the occurrence of event $B$.
Multiplication Rule for Independent Events:
$$P(A \cap B) = P(A) \times P(B)$$Two events $A$ and $B$ are dependent if the occurrence of event $A$ affects the probability of the occurrence of event $B$ (e.g., drawing items without replacement).
Multiplication Rule for Dependent Events:
$$P(A \cap B) = P(A) \times P(B \text{ given } A)$$Two events $A$ and $B$ are mutually exclusive if they cannot occur at the same time. They have no common outcomes ($A \cap B = \emptyset$).
Addition Rule for Mutually Exclusive Events:
$$P(A \cup B) = P(A) + P(B)$$Two events $A$ and $B$ are non-mutually exclusive if they can occur simultaneously. They share common outcomes ($A \cap B \ne \emptyset$).
Addition Rule for Non-Mutually Exclusive Events:
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$The probability of the event "not $A \cup B$" is given by:
$$P((A \cup B)') = 1 - P(A \cup B)$$| Event Relationship | Key Formula | Keywords / Indicator |
|---|---|---|
| Independent ($A$ and $B$) | $P(A \cap B) = P(A) \times P(B)$ | "and", with replacement |
| Dependent ($A$ and $B$) | $P(A \cap B) = P(A) \times P(B|A)$ | "and", without replacement |
| Mutually Exclusive ($A$ or $B$) | $P(A \cup B) = P(A) + P(B)$ | "or", no overlap |
| Non-Mutually Exclusive ($A$ or $B$) | $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ | "or", shared outcomes exist |