Unit 4: Foundations of Probability

4.1 Basic Terminology

Random Experiment

An experiment or process whose outcome cannot be predicted with certainty, but all possible outcomes are known.

• Example 1: Tossing a coin.
• Example 2: Rolling a die.
• Example 3: Measuring the lifetime of a light bulb.

Sample Point

A single possible outcome of a random experiment.

• Example (Rolling a die): The number '4' is a sample point.

Sample Space (S)

The set of all possible outcomes (all sample points) of a random experiment.

• Example 1 (Tossing one coin): S = {Head, Tail}
• Example 2 (Rolling one die): S = {1, 2, 3, 4, 5, 6}
• Example 3 (Tossing two coins): S = {HH, HT, TH, TT}

4.2 Events and Algebra of Events

Event

An event is any subset of the sample space S. It is a collection of one or more sample points.

• Example (Rolling a die, S = {1, 2, 3, 4, 5, 6}):
  • Let A be the event "getting an even number." Then A = {2, 4, 6}.
  • Let B be the event "getting a number greater than 4." Then B = {5, 6}.

Types of Events

• Simple Event: An event with only one sample point (e.g., A = {1}).
• Compound Event: An event with more than one sample point (e.g., A = {2, 4, 6}).
• Sure Event (or Certain Event): The entire sample space S. (e.g., "getting a number from 1 to 6"). P(S) = 1.
• Impossible Event (or Null Event): The empty set (Ø). (e.g., "getting a 7"). P(Ø) = 0.

Algebra of Events (Set Operations)

Since events are sets, we can use set theory to combine them.

[Image of Venn diagrams illustrating A union B, A intersection B, and A complement]

Operation	Notation	Meaning ("In Words")	Example (A={1,2}, B={2,3})
Union	A ∪ B	"Event A OR B or Both"	{1, 2, 3}
Intersection	A ∩ B	"Event A AND B"	{2}
Complement	A' or Aᶜ or A-bar	"Event NOT A"	If S={1,2,3,4}, A' = {3, 4}