Summary

14.2 Summary

Description

Quick Overview

This section outlines key definitions and concepts related to theoretical probability.

Standard

In this section, we explore the foundational principles of theoretical probability, including definitions of certain, impossible, and complementary events, as well as the constraints of probability values. It establishes the groundwork for further studies in probability theory.

Detailed

Detailed Summary

In this section, we delve into the foundational definitions of theoretical probability, crucial for understanding probability theory as a whole. The theoretical probability of an event E is defined by the formula:

P(E) = Number of outcomes favourable to E / Number of all possible outcomes of the experiment,

assuming that all outcomes are equally likely. We categorize events into several types:
1. Sure Event: Probability is 1, meaning the event is guaranteed to occur.
2. Impossible Event: Probability is 0, indicating that the event cannot occur under any circumstances.
3. The probability of any event E falls within the bounds of 0 and 1 inclusive: 0 ≤ P(E) ≤ 1.
4. In probability theory, an event that has only one outcome is referred to as an elementary event, and the sum of the probabilities of all elementary events adds up to 1.
5. Furthermore, for any event E, the sum of probabilities of E and its complement (not E) is also 1, establishing the property of complementary events.

This section serves as a critical introduction to theoretical probability, paving the way for deeper exploration in subsequent chapters.

Key Concepts

  • Theoretical Probability: Calculated based on equally likely outcomes, defined as P(E) = Favorable outcomes / Total outcomes.

  • Sure Event: Probability of an event that will definitely occur, equals 1.

  • Impossible Event: Probability of an event that cannot occur, equals 0.

  • Elementary Event: An event with only one outcome.

  • Complementary Events: Two events that cannot happen at the same time, where P(E) + P(not E) = 1.

Memory Aids

🎵 Rhymes Time

  • Probability goes from zero to one; Impossible events are no fun!

📖 Fascinating Stories

  • In a game, you either win or lose. The probability of winning is a number you choose.

🧠 Other Memory Gems

  • Use 'SIP' to remember: Sure = 1, Impossible = 0, and Probability lies in between!

🎯 Super Acronyms

Think 'ECP' for Event, Complement, and Probability.

Examples

  • Calculating the probability of rolling a die and getting a number 5: There is 1 favorable outcome (5) out of 6 possible outcomes; hence P(5) = 1/6.

  • If a deck of cards is shuffled, the probability of drawing an Ace (4 favorable outcomes out of 52 possible outcomes) is 4/52, which simplifies to 1/13.

Glossary of Terms

  • Term: Theoretical Probability

    Definition:

    The probability calculated based on the assumption of equally likely outcomes.

  • Term: Sure Event

    Definition:

    An event that is certain to happen, with a probability of 1.

  • Term: Impossible Event

    Definition:

    An event that cannot happen, with a probability of 0.

  • Term: Elementary Event

    Definition:

    An event containing only one outcome.

  • Term: Complementary Events

    Definition:

    Two events where the occurrence of one event means the other cannot occur.