Combinations of Events & Complementary Events
1. Review: Events and Sample Space
Before diving into combinations of events, let's recall the foundational concepts:
- Sample Space ($S$): The set of all possible outcomes of an experiment.
- Event ($A$): A subset of the sample space — a collection of outcomes we're interested in.
- Probability of an event: $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$, when all outcomes are equally likely.
2. Combinations of Events
We can combine events using set operations to create new, more complex events. There are three fundamental combinations:
Union: $A \cup B$
"A or B (or both)"
The event that at least one of $A$ or $B$ occurs.
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
This is the Addition Rule.
Intersection: $A \cap B$
"A and B"
The event that both $A$ and $B$ occur simultaneously.
$$P(A \cap B) = P(A) \cdot P(B|A)$$
This is the Multiplication Rule.
Complement: $\bar{A}$ or $A'$
"Not A"
The event that $A$ does not occur.
$$P(\bar{A}) = 1 - P(A)$$
This is the Complement Rule.