Lecture Notes on Probability

Introduction to Probability

• Definition: Probability is a measure of the likelihood that an event will occur.
• Applications: Used in various fields such as statistics, finance, gambling, science, and engineering.

Key Concepts

Basic Terms

• Experiment: A process that leads to the occurrence of one and only one of several possible observations.
• Outcome: A possible result of an experiment.
• Event: A specific outcome or a set of outcomes of a random experiment.

Probability Principles

• Sample Space (S): The set of all possible outcomes of an experiment.
• Event (E): A subset of the sample space.
• Probability of an Event (P(E)): Number of favorable outcomes divided by the total number of possible outcomes.
  • Formula: ( P(E) = \frac{n(E)}{n(S)} )

Types of Probability

• Classical Probability: Based on the assumption that outcomes of an experiment are equally likely.
• Empirical Probability: Based on observations and experiments.
• Subjective Probability: Based on intuition or personal judgment.

Important Properties

• The probability of any event is a number between 0 and 1.
• The sum of probabilities of all mutually exclusive outcomes of an experiment is 1.
• The probability of the complement of an event is 1 minus the probability of the event.
  • Formula: ( P(E') = 1 - P(E) )

Conclusion

• Understanding probability is essential for making predictions about random events.
• Probability forms the foundation of statistics and is critical in various decision-making processes.