Probability Lecture on Counting Techniques

Overview

• Chapter 15: Probability
  • Weightage: 10 marks
  • 425 marks from this chapter.

Key Topics

• Counting the Number of Possible Outcomes
  • Three main topics: Fundamental Principle, Permutation, Combination.
• Simple Probability
• Learning Outcomes:
  • Understanding probability distribution
  • Using counting techniques like the counting rule (M * factorial).

Fundamental Principle (Multiplication Rule)

• Main Rule: If there are M ways of doing one thing and N ways of doing another, and the two actions are independent, then there are M * N ways of doing both.
• Example 1:
  • 10 pairs of shoes: Possible outcomes = n1 * N2 = 120.
• Example 2:
  • Car manufacturer sells models with options:
    • Body style: 3 types.
    • Leather types: 3 types.
    • Fabric colors: 6 colors.
  • Total possible outcomes = 3 * 3 * 6 = 54.

Additional Examples

• Three-digit numbers using digits (2, 3, 4, 5, 7):
  • Without repetition (unique digits): n1 = 5, n2 = 4, n3 = 3
  • With repetition: n1 = n2 = n3 = 5
• Three-letter combinations (not repeating):
  • Given seven letters, form three letters combinations: n1 * n2 * n3 = 7 * 6 * 5 = 210.
  • With repetition: n1 = n2 = n3 = 7.

Special Cases in Counting (Zero not countable)

• Four-digit numbers using digits (0, 1, 2, 3, 5, 7):
  • Zero not countable in the first position.
  • Without repetition: First digit (5 choices), next digits (6 choices each).
  • Repeated digits count: First digit still cannot be zero.
    • Example: n1 = 5, n2 = n3 = n4 = 6.

Odd & Even Numbers

• Generating odd/even numbers with given digits.
  • Example of forming even and odd numbers:
    • Identify numbers ending in odd/even digits
    • Calculate number of samples and identify valid formations.

Practice Questions

• Generate 2-digit odd numbers from given digits.
  • Identify odd numbers in the unit position.
  • Find all possible combinations and check for odd/even validity.