HSSLIVE Plus One Maths Chapter 16: Probability Notes

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Probability theory quantifies uncertainty and provides a framework for analyzing random phenomena. This chapter introduces probability axioms, conditional probability, independence, random variables, and probability distributions. Students will develop skills in calculating probabilities of complex events, understanding expectation and variance, and applying probability models to real-world scenarios. This mathematical foundation is essential for statistics, risk assessment, quality control, actuarial science, machine learning, and many other fields where decisions must be made under uncertainty.

Chapter 16: Probability

Basic Concepts:

  • Experiment: An operation with well-defined outcomes
  • Sample Space (S): Set of all possible outcomes
  • Event: A subset of the sample space

Types of Events:

  • Simple Event: Contains only one outcome
  • Compound Event: Contains more than one outcome
  • Impossible Event: Contains no outcomes (empty set)
  • Sure Event: Contains all outcomes (entire sample space)

Probability Definitions:

  1. Classical Definition: P(E) = Number of favorable outcomes / Total number of possible outcomes
  2. Axiomatic Approach:
    • P(E) ≥ 0 for any event E
    • P(S) = 1 where S is the sample space
    • For mutually exclusive events E and F, P(E ∪ F) = P(E) + P(F)

Probability Laws:

  1. Addition Law:
    • P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
    • For mutually exclusive events: P(A ∪ B) = P(A) + P(B)
  2. Conditional Probability: P(A|B) = P(A ∩ B)/P(B), where P(B) > 0
  3. Multiplication Law:
    • P(A ∩ B) = P(A) × P(B|A)
    • For independent events: P(A ∩ B) = P(A) × P(B)

Bayes’ Theorem:

P(A|B) = [P(B|A) × P(A)]/P(B)

Random Variables:

A function that assigns a real number to each outcome in the sample space.

  • Probability Mass Function (PMF): For discrete random variables
  • Probability Density Function (PDF): For continuous random variables

Expected Value:

For a discrete random variable X with values x₁, x₂, …, xₙ and probabilities p₁, p₂, …, pₙ: E(X) = Σxipi

Complete Chapter-wise Hsslive Plus One Maths Notes

Our HSSLive Plus One Maths Notes cover all chapters with key focus areas to help you organize your study effectively:

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