Introduction to Probability

Sample Space

The sample space is the set of all possible outcomes of a statistical experiment. It can be finite or infinite, and each outcome is called an element or member of the sample space.

• Definition: The sample space, denoted by S, includes every possible result of an experiment.

• Example 1: Tossing a die:

  • If we are interested in the number that shows on the top face,

  • If we are interested in whether the number is even or odd,

• Example 2: Flipping a coin until a tail appears:

Events

An event is a subset of a sample space. It can include the entire sample space, the null set (denoted by ), or any subset of outcomes.

• Example: For :

  • Event A = {even number} = {2,4,6}

  • Event B = {prime number} = {2,3,5}

  • Event C = {number greater than 6} =

  • Event D = {number less than 7} = S

Operations on Events

• Intersection: is the set of elements common to both A and B. Example: If and , , then .

• Union: is the set of elements belonging to A or B or both. Example:

• Mutually Exclusive Events: and are mutually exclusive if .

• Partition of Sample Space: Events form a partition if they are mutually exclusive and .

Venn Diagrams

Venn diagrams are used to visually represent the relationships between events and the sample space.

Counting Principles

Fundamental Principle of Counting

If one operation can be performed in ways and another in ways, then both can be performed in ways.

Arrangement Without Repetition

The number of ways to choose objects in order out of distinct objects is:

• Remark: Order matters, no repetition.

• Example: Assigning awards to students from a class of 25.

Combination

The number of ways to choose objects out of distinct objects where order does not matter is:

• Remark: No order, no repetition.

Permutation

A permutation is an ordered arrangement of a set of elements.

• Number of permutations of objects:

• Example: Number of ways to arrange 5 trees in a row:

Circle Permutation

Number of permutations of distinct objects in a circle:

Permutation of Sets Containing Similar Objects

For objects, with of one kind, of another, ..., of a th kind:

• Example: Arranging players in a football team with different positions.

Partitioning into Sets

Number of ways to partition objects into cells with elements:

Probability Concepts

Definition of Probability

Probability is a measure of the likelihood that an event will occur, ranging from 0 (impossible) to 1 (certain).

Complementary Events

If and are complementary, then:

Classical Probability Formula

If all outcomes are equally likely:

• where = number of favorable outcomes, = number of possible outcomes

Additive Rules

For events and :

• If are mutually exclusive:

• If partition :

• For three events :

Conditional Probability

The probability of event given event has occurred:

Independent Events

Events and are independent if:

Multiplicative Rule

If both and occur:

Bayes' Rule

Bayes' theorem allows us to update probabilities based on new information:

Applications and Examples

• Calculating probabilities in coin tosses, card hands, and selection problems.

• Using probability rules to solve real-world problems, such as reliability of machines, selection of students, and quality control in manufacturing.

Table: Categorization of Adults in a Small Town

This table classifies adults by gender and employment status.

Main purpose: Classification and calculation of probabilities based on employment and gender.

Summary of Key Formulas

• Number of arrangements (permutations):

• Number of combinations:

• Probability of event:

• Additive rule:

• Conditional probability:

• Multiplicative rule:

• Bayes' rule:

Additional info: These notes cover foundational probability concepts, including sample spaces, events, counting principles, probability rules, and applications. They are suitable for introductory college-level physics or statistics courses.