Fundamentals of Probability and Basic Statistics

Introduction to Probabilities

Definition of Probability

Probability is the chance that a given event will occur. It is a measure that quantifies the likelihood of outcomes in a random experiment, expressed as a number between 0 and 1, or equivalently, between 0% and 100%.

• Probability Formula: $P(A) = \frac{\#A}{\text{TOTAL NUMBER}}$ where $\#A$ is the number of ways event A can occur, and TOTAL NUMBER is the total number of possible outcomes.

• Sample Space (S): The set of all possible outcomes of an experiment. Example: Flipping a fair coin: $S = \{\text{Heads}, \text{Tails}\}$

• Probability Values: Probability values fall between 0 (impossible event) and 1 (certain event), or 0% and 100%.

Basic Examples

• Flipping a Coin: $S = \{\text{Heads}, \text{Tails}\}$ Probability of getting heads: $P(\text{Heads}) = \frac{1}{2} = 0.5$

• Rolling a Fair Six-Sided Die: $S = \{1,2,3,4,5,6\}$ Probability of getting an even number: $P(\text{Even}) = \frac{3}{6} = 0.5$

• Drawing a Card: $S = \{\text{Hearts}, \text{Clubs}, \text{Spades}, \text{Diamonds}\}$ Probability of drawing a red card: $P(\text{Red}) = \frac{2}{4} = 0.5$

Types of Probability

Empirical Probability

Empirical probabilities are based on outcomes of an experiment, calculated by counting the number of times something occurred.

• Formula: $P(E) = \frac{\text{Number of times event E occurs}}{\text{Total number of trials}}$

• Example: If 4 people flip a coin and 1 gets heads, $P(\text{Heads}) = \frac{1}{4} = 0.25$

Theoretical Probability

Theoretical probabilities are based on the understanding of a process in theory.

• Example: Probability of flipping a coin and getting heads: $P(\text{Heads}) = \frac{1}{2} = 0.5$

The Number in the Denominator Matters

• Possible probabilities: $0/4 = 0\%$, $1/4 = 25\%$, $2/4 = 50\%$, $3/4 = 75\%$, $4/4 = 100\%$

• Empirical probability approaches theoretical probability as the number of trials increases.

The Law of Large Numbers

Explanation

The Law of Large Numbers states that as the number of trials increases, the empirical probability of a particular outcome approaches the theoretical probability in the long run.

• Example: Flipping a fair coin many times, the proportion of heads will approach 0.5.

Simple Probabilities and Complements

Complements

The complement of an event A is the event that A does not occur.

• Formula: $P(\text{not } A) = 1 - P(A)$

• Example: Probability of not rolling a 4 on a six-sided die: $P(\text{not } 4) = 1 - P(4) = 1 - \frac{1}{6} = \frac{5}{6}$

Intersections and Unions

Intersections

The intersection of two events A and B (denoted $A \cap B$) is the event that both A and B occur.

• Formula: $P(A \cap B) = \frac{\text{ways to get both event A and B}}{\text{total possible outcomes}}$

• Keyword: "and"

Unions

The union of two events A and B (denoted $A \cup B$) is the event that either A or B or both occur.

• Formula: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Keyword: "or"

Venn Diagram

A Venn diagram visually represents the relationships between events. The intersection (A and B) is shown as the overlapping area.

Conditional Probability

Definition

Conditional probability is the probability of event A occurring given that event B has occurred.

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

• Keyword: "given"

• The denominator is the probability of the given event.

Example

• Probability of rolling an even number greater than 2 on a six-sided die: Even numbers: 2, 4, 6; Numbers greater than 2: 3, 4, 5, 6 Even numbers > 2: 4, 6 $P(\text{even} | >2) = \frac{2}{4} = 0.5$

Contingency Tables

Purpose

Contingency tables are used to organize data for calculating probabilities involving two characteristics.

Eye Color	Male	Female	Total
Green	28	44	72
Not Green	406	353	759
Total	434	397	831

• Probability of green eyes: $P(\text{Green}) = \frac{72}{831} = 0.09$

• Probability of male: $P(\text{Male}) = \frac{434}{831} = 0.52$

• Probability of green eyes and male: $P(\text{Green and Male}) = \frac{28}{831} = 0.03$

• Probability of green eyes or male: $P(\text{Green or Male}) = \frac{72 + 434 - 28}{831} = 0.55$

Independence

Definition

Two events are independent if the occurrence of one does not affect the chance of the occurrence of the other.

• Example: Flipping a coin twice: Probability of getting heads in the second flip given the first flip is heads: $P(\text{H in 2nd flip} | \text{H in 1st flip}) = \frac{1}{2} = 0.5$

Summary Table: Probability Formulas

Concept	Formula
Simple Probability	$P(A) = \frac{\#A}{\text{Total outcomes}}$
Complement	$P(\text{not } A) = 1 - P(A)$
Intersection	$P(A \cap B) = \frac{\text{ways to get both A and B}}{\text{Total outcomes}}$
Union	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$
Conditional Probability	$P(A|B) = \frac{P(A \cap B)}{P(B)}$
Empirical Probability	$P(E) = \frac{\text{Number of times E occurs}}{\text{Total trials}}$

Additional info:

• These notes cover foundational probability concepts essential for introductory statistics, including empirical and theoretical probability, complements, intersections, unions, conditional probability, and independence.

• Examples and tables are based on typical classroom scenarios and standard probability experiments.