Overview
This lecture introduces probability, its foundational concepts, and key types of probability distributions, using real-world examples like movie choices and birdwatching.
Introduction to Probability
- Probability quantifies uncertainty and helps predict how likely different outcomes are.
- All possible outcomes of a situation form the sample space (e.g., comedy, drama, horror movies).
- Probability originated in studying games of chance and insurance risks.
Classical vs. Empirical Probability
- Classical probability assumes all outcomes are equally likely (e.g., 1/3 chance for each movie type if all are shown equally).
- Empirical probability uses past data to estimate chances when outcomes may not be equally likely.
- A random sample can be used when all historical data isn’t available, by observing proportions in the sample.
Events and Probability Distributions
- An event is a collection of one or more outcomes (e.g., seeing either a comedy or drama).
- Probability of an event sums the probabilities of all included outcomes.
- Probability distributions describe how likely different outcomes are for a scenario.
Types of Probability Distributions
- Discrete probability distributions handle outcomes that are countable (e.g., number of movies).
- Probability Mass Functions (PMFs) assign probabilities to each discrete outcome.
- Continuous probability distributions describe outcomes that can take any value within a range (e.g., length of movie previews).
- Probability Density Functions (PDFs) assign probabilities to intervals, and area under the curve represents probability.
Common Probability Distributions
- Uniform distribution: all outcomes are equally likely (e.g., finding a bird anywhere in a region).
- Binomial distribution: only two outcomes are possible (e.g., seeing a male or female condor).
- Exponential distribution: models the time between events (e.g., waiting time between bird sightings).
Properties of Probability
- Probability values range from 0 (impossible) to 1 (certain).
- The total probability across all possible outcomes equals 1.
- Probabilities outside this range indicate an error in calculation.
Key Terms & Definitions
- Sample Space — The set of all possible outcomes in a scenario.
- Classical Probability — Probability calculated assuming all outcomes are equally likely.
- Empirical Probability — Probability estimated based on observed data.
- Event — A set of one or more outcomes of interest.
- Probability Mass Function (PMF) — Describes probabilities for discrete outcomes.
- Probability Density Function (PDF) — Describes probabilities for continuous outcomes.
- Uniform Distribution — All outcomes have the same probability.
- Binomial Distribution — Probability distribution with two possible outcomes (success/failure).
- Exponential Distribution — Describes the time between random events.
Action Items / Next Steps
- Review empirical probability by collecting and analyzing a random sample from real-world data.
- Practice identifying which probability distribution fits different scenarios.
- Read about PMFs and PDFs to reinforce understanding of discrete and continuous distributions.