Probability Theory: Sample Spaces and Event Types

1. Sample Space: Binary Signal Example

Definition: A sample space (S) is the set of all possible outcomes of a random experiment, usually denoted by S.

Example: In digital communication, a binary signal has only two possible values: 0 and 1. Hence, the sample space is S = {0, 1}. This means every outcome of the experiment must belong to this set.

2. Event Definitions and Types

Definition: An event (E) is any subset of the sample space, representing one or more outcomes of an experiment.

Types:

• Simple Event: Contains only one outcome. Example: E = {0}
• Compound Event: Contains more than one outcome. Example: E = {0, 1}
• Impossible Event: An event that cannot occur, represented by the empty set (∅). Example: Getting a ‘2’ in a binary system.

3. Deterministic vs. Random Experiments

A deterministic experiment is one in which the outcome is fixed and can be predicted with certainty. There is no randomness involved. Example: Switching ON a light → it glows.

A random experiment is one in which the outcome cannot be predicted in advance and may vary each time. Example: Tossing a coin → head or tail.

Thus, deterministic experiments are predictable, while random experiments involve uncertainty.

4. Mutually Exclusive and Exhaustive Events

Mutually Exclusive Events: These are events that cannot occur at the same time. If one event happens, the other cannot.

• Condition: A ∩ B = ∅
• Example: In a binary system, if event A = {0} and B = {1}, both cannot occur together.

Exhaustive Events: These are events that, together, include all possible outcomes of the sample space.

• Condition: A ∪ B = S
• Example: If S = {0, 1}, A = {0}, and B = {1}, together they cover the whole sample space.

5. Complement of an Event

Definition: The complement of an event A is the set of all outcomes in the sample space that are not in A. It is denoted by A′.

Formula: A′ = S - A

Example: If S = {0, 1} and A = {0}, then A′ = {1}. This means if event A does not occur, its complement must occur.

6. Venn Diagram Representation

A Venn diagram is a graphical representation used to show relationships between events:

• The rectangle represents the sample space (S).
• Circles inside the rectangle represent events (A, B).
• The overlapping region shows the intersection (A ∩ B), representing common outcomes.
• The combined area of circles shows the union (A ∪ B).
• The area outside a circle represents the complement (A′).

Venn diagrams help in visualizing operations like union, intersection, and complement of events clearly.