L16

Sep 20, 2024

Lecture Notes: Probability and Counting Principles

Recap of Probability Concepts

  • Sample Space: List of all possible outcomes.
  • Event: A subset of the sample space. The number of possibilities in an event is less than or equal to those in the sample space.

Example: Coin Toss

  • Single Coin Toss: Sample space = {H, T}
  • Two Successive Coin Tosses: Sample space = {HH, HT, TH, TT}
  • As the number of coin tosses increases, the sample space increases.

Axioms of Probability

  1. Probability of Event: Bounded between 0 and 1.
    • Example: Event E with no outcomes (null event) has a probability of 0.
    • Event E as the sample space itself has a probability of 1.
    • Event E as a head appearing has a probability of 0.5.

Venn Diagrams and Events

  • Used to describe events and their intersection.
  • Mutually Exclusive Events: No overlap, P(E1 ∩ E2) = 0.
  • Union of Events: P(E1 ∪ E2) = P(E1) + P(E2) - P(E1 ∩ E2).

Counting Principles

Example: Balls in a Box

  • Setup: 5 blue balls, 3 red balls.
  • Question: Probability of drawing 2 red and 1 blue ball.

Permutations and Combinations

  • Permutations: Ordering matters.

    • Example: Arrange a, b, c → 6 possibilities (ABC, ACB, BAC, BCA, CAB, CBA).
    • Formula: n!/(n-r)! for arranging r objects from n.

  • Combinations: Order does not matter.

    • Formula: nCr = n! / (r!(n-r)!)
    • Example: Committee of 2 girls and 1 boy from 10 students: 4C2 * 6C1 / 10C3.

Factorials

  • n!: Product of all positive integers up to n.
  • Examples:
    • 1! = 1
    • 2! = 2
    • 3! = 6
    • 4! = 24

Examples in Context

  • Books on a Shelf: Arranging books by category.

    • Categories can be ordered in factorial ways.
    • Arrangement within each category also follows factorial rules.

  • Committee Selection: Probability of selecting specific members.

    • Example: Always selecting a member 'a' in k selections from n items: (n-1)C(k-1) / nCk.

Summary

  • Probability Basics: Revisiting the core principles and examples.
  • Counting Techniques:
    • Permutation: Used where order matters.
    • Combination: Used where selection is key, order does not matter.

These notes outline the key concepts of probability, permutations, and combinations covered in the lecture, providing a basis for understanding how to calculate probabilities and count outcomes effectively.