Probability Basics

Probability

Definition: The study of randomness and uncertainty; quantifies how likely events are to occur.
Key Concepts:
- Experiment: A procedure that yields one of a possible set of outcomes.
- Sample Space (S): The set of all possible outcomes of an experiment.
- Event: A subset of the sample space; can consist of one or more outcomes.
- Probability of an Event (P): A measure of the likelihood of the event occurring, calculated as:
  - P(Event) = (Number of favorable outcomes) / (Total number of outcomes)
Types of Probability:
- Theoretical Probability: Based on reasoning or logical analysis.
- Experimental Probability: Based on the results of experiments or trials.
- Subjective Probability: Based on personal judgment or experience.
Rules of Probability:
- Addition Rule: For two mutually exclusive events A and B:
  - P(A or B) = P(A) + P(B)
- Multiplication Rule: For independent events A and B:
  - P(A and B) = P(A) * P(B)

Statistics

Definition: The science of collecting, analyzing, interpreting, presenting, and organizing data.
Key Concepts:
- Descriptive Statistics: Summarizes and describes the features of a dataset.
  - Measures of Central Tendency: Mean, median, mode.
  - Measures of Dispersion: Range, variance, standard deviation.
- Inferential Statistics: Makes predictions or inferences about a population based on a sample.
  - Hypothesis Testing: Procedure to test assumptions regarding a population parameter.
  - Confidence Intervals: Range of values used to estimate the true parameter.
Data Types:
- Qualitative (Categorical): Non-numeric data, e.g., gender, color.
- Quantitative (Numeric): Numeric data, can be discrete (countable) or continuous (measurable).
Common Distributions:
- Normal Distribution: Bell-shaped curve; characterized by mean and standard deviation.
- Binomial Distribution: Models the number of successes in a fixed number of independent trials.
- Poisson Distribution: Models the number of events occurring within a fixed interval of time or space.
Correlation and Regression:
- Correlation: Measures the strength and direction of a linear relationship between two variables.
  - Correlation Coefficient (r): Ranges from -1 to 1.
- Regression: Predicts the value of a dependent variable based on the value of one or more independent variables.
  - Linear regression formula: Y = a + bX, where a is the y-intercept and b is the slope.
Common Statistical Tests:
- T-test: Compares means between two groups.
- Chi-square Test: Assesses relationships between categorical variables.
- ANOVA (Analysis of Variance): Compares means among three or more groups.

Probability

Study of randomness and uncertainty, quantifying event likelihood.
Experiment: Procedure yielding one or more outcomes.
Sample Space (S): Complete set of all possible outcomes from an experiment.
Event: Subset of the sample space, can include multiple outcomes.
Probability of an Event (P): Computed as P(Event) = (Number of favorable outcomes) / (Total number of outcomes).
Types of Probability:
- Theoretical Probability: Derived from logical reasoning.
- Experimental Probability: Based on outcomes from actual experiments or trials.
- Subjective Probability: Based on personal intuition or experience.
Rules of Probability:
- Addition Rule: For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
- Multiplication Rule: For independent events A and B, P(A and B) = P(A) * P(B).

Statistics

Science dedicated to collecting, analyzing, interpreting, presenting, and organizing data.
Descriptive Statistics: Summarizes dataset features.
- Measures of Central Tendency: Mean, median, mode to characterize data centrality.
- Measures of Dispersion: Range, variance, standard deviation reflect data spread.
Inferential Statistics: Draws conclusions about a population based on sample data.
- Hypothesis Testing: Procedure for testing assumptions of population parameters.
- Confidence Intervals: Estimate range for a population parameter.
Data Types:
- Qualitative (Categorical): Non-numeric data like gender and color.
- Quantitative (Numeric): Numeric data; can be either discrete or continuous.
Common Distributions:
- Normal Distribution: Characterized by a bell shape, defined by mean and standard deviation.
- Binomial Distribution: Represents the number of successes in a set number of trials.
- Poisson Distribution: Models event occurrences over a specified interval.
Correlation and Regression:
- Correlation: Strength and direction of the linear relationship between two variables; measured by the correlation coefficient (r) ranging from -1 to 1.
- Regression: Used to predict dependent variable values based on independent variables; linear regression formula Y = a + bX, where 'a' is the y-intercept and 'b' the slope.
Common Statistical Tests:
- T-test: Compares means between two groups.
- Chi-square Test: Evaluates relationships between categorical variables.
- ANOVA (Analysis of Variance): Compares means across three or more groups.