Fundamentals of Probability and Events in Statistics

Probability Basics

Definition of Probability

Probability is a measure of uncertainty, quantifying the likelihood that a particular event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

• Probability Model: A mathematical description of an experiment based on its possible outcomes and their associated probabilities.

• Experiment: An action whose outcome cannot be predicted with certainty.

• Event: A specified result that may or may not occur when an experiment is performed.

Equal-Likelihood Model

When all outcomes of an experiment are equally likely, the probability of an event is calculated as:

• Formula: $P(E) = \frac{\text{Number of outcomes in } E}{\text{Total number of possible outcomes}}$

• Example: If a simple random sample (SRS) of 2 is chosen from Abby, Leah, Rebecca, and Joshua, the probability that Leah and Rebecca are chosen is: $P(\text{Leah and Rebecca}) = \frac{1}{6}$

Basic Properties of Probabilities

Probability Properties

• Property 1: The probability of any event is always between 0 and 1.

• Property 2: The probability of an event that cannot occur is 0 (impossible event).

• Property 3: The probability of an event that must occur is 1 (certain event).

Example: Rolling Dice

Consider rolling two dice. The probability that the sum is 6:

• Possible outcomes for sum of 6: (1,5), (2,4), (3,3), (4,2), (5,1)

• Total possible outcomes: 36

• Formula: $P(\text{sum is 6}) = \frac{5}{36}$

Sample Space and Events

Definitions

• Sample Space (S): The collection of all possible outcomes for an experiment.

• Event: Any subset of the sample space. An event occurs if the outcome of the experiment is in the event.

Example: Family Selection

A U.S. family is selected at random. The sample space may include: married, single, divorced, etc. An event could be "family has three persons".

Relationships Among Events

Types of Events

• Complement (Ec): The event "E does not occur".

• Intersection (A ∩ B): The event "both A and B occur".

• Union (A ∪ B): The event "either A or B or both occur".

Venn Diagrams

Venn diagrams are used to visually represent relationships among events, such as intersection, union, and complement.

Mutually Exclusive Events

• Two or more events are mutually exclusive if they have no outcomes in common.

• Example: Rolling a die, the events "rolling a 3" and "rolling a 5" are mutually exclusive.

Some Rules of Probability

Probability Notation

• P(E): Probability that event E occurs.

• Complement Rule: $P(E^c) = 1 - P(E)$

Special Addition Rule

• If A and B are mutually exclusive, then: $P(A \cup B) = P(A) + P(B)$

Example: Rolling a Die

• Let A = rolling a 2, B = rolling a 5. $P(A \cup B) = P(A) + P(B) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6}$

Contingency Tables: Joint and Marginal Probabilities

Contingency Table Example

Contingency tables display the frequency distribution of variables and allow calculation of joint and marginal probabilities.

Income	Not Happy	Pretty Happy	Very Happy	Total
Above Average	31	241	140	412
Average	94	1207	754	2055
Below Average	86	423	40	549
Total	211	1871	934	3016

• Marginal Probability: Probability of a single variable (e.g., probability of being "Very Happy").

• Joint Probability: Probability of two variables occurring together (e.g., probability of being "Very Happy" and having "Above Average" income).

Converting to Joint Probability Distribution

Divide each cell frequency by the total number of observations to obtain joint probabilities.

Income	Not Happy	Pretty Happy	Very Happy	Total
Above Average	0.010	0.080	0.046	0.137
Average	0.031	0.400	0.250	0.681
Below Average	0.029	0.140	0.013	0.182
Total	0.070	0.620	0.309	1.000

Conditional Probability

Definition

The probability that event B occurs given that event A occurs is called a conditional probability, denoted as $P(B|A)$.

• Formula: $P(B|A) = \frac{P(A \cap B)}{P(A)}$

• Example: Probability that an American is very happy given they have above average income: $P(\text{Very Happy} | \text{Above Average}) = \frac{140}{412} \approx 0.34$

Multiplication Rule and Independence

Multiplication Rule

• If A and B are independent events: $P(A \cap B) = P(A) \times P(B)$

Example: Tossing a Coin

• Let A = first toss is heads, B = second toss is heads. $P(A \cap B) = P(A) \times P(B) = 0.5 \times 0.5 = 0.25$

Tree Diagrams and Compound Probability

Tree Diagrams

Tree diagrams are useful for visualizing compound events and calculating probabilities for sequential experiments.

• Example: Selecting two students from a class with 7 boys and 5 girls. The probability that both are girls: $P(\text{both girls}) = \frac{5}{12} \times \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$

Univariate and Bivariate Data

Definitions

• Univariate Data: Data from one variable of a population.

• Bivariate Data: Data from two variables of a population.

Frequency distributions are used to group univariate data, while contingency tables are used for bivariate data.

Summary Table: Key Probability Concepts

Concept	Definition	Formula
Probability	Measure of likelihood of an event	$P(E) = \frac{\text{Number of outcomes in } E}{\text{Total outcomes}}$
Complement	Event does not occur	$P(E^c) = 1 - P(E)$
Mutually Exclusive	Events with no outcomes in common	$P(A \cap B) = 0$
Conditional Probability	Probability of B given A	$P(B|A) = \frac{P(A \cap B)}{P(A)}$
Multiplication Rule	Probability of A and B (independent)	$P(A \cap B) = P(A) \times P(B)$

Additional info: Some context and examples have been expanded for clarity and completeness, including definitions and formulas for key probability concepts.