Lecture on Probability: Permutation, Combination, and Conditional Probability

Recap of Previous Lectures

• Sample and Sample Space: List of all possible outcomes.
• Event: Corresponds to a particular outcome or outcomes.
• Probability Definitions:
  • Probability of sample space = 1
  • Probability of an event E is between 0 and 1.
  • If Event E = S (sample space), then probability of E = 1.

Venn Diagrams in Probability

• Representation: Event E as a subset of a sample space S.
• Complementary Event:
  • E complement (E^C) = the area where E is not present.
  • Probability relation: P(E) + P(E^C) = 1.
  • Example: If E = girl, then E^C = boy.

Permutation and Combination

• Definitions:
  • Permutation (nPr): Arrangement with order. Formula: ( nPr = \frac{n!}{(n-r)!} )
  • Combination (nCr): Selection without order. Formula: ( nCr = \frac{n!}{r!(n-r)!} )
• Examples:
  • Committee Selection: From 5 men and 5 women, select a committee of 4. Probability of having 3 women and 1 man.
    • Total ways: 10C4
    • 3 women from 5: 5C3
    • 1 man from 5: 5C1
    • Probability: ( \frac{5C3 \times 5C1}{10C4} )

Birthday Problem

• Scenario: Probability that no two people share the same birthday among n people.
• Total Possibilities: 365^n (non-leap year assumed).
• Calculation:
  • Person 1: 365 days
  • Person 2: 364 days, etc.
  • Probability no shared birthday: ( \frac{365 \times 364 \times ...}{365^n} )
• Conclusion: Probability decreases as n increases; for n = 23, probability ≈ 50%.

Introduction to Conditional Probability

• Example: Rolling a die twice.
  • Sample space: 6 x 6 = 36 possibilities.
  • Probability x + y = 8: Possibilities are (2,6), (3,5), (4,4), (5,3), (6,2).
  • Conditional probability: Given first roll is 3, find probability of total 8.
  • Reduced sample space: 6 possibilities
  • Probability: 1/6 (only (3,5) works).

Venn Diagram and Conditional Probability

• Concepts:
  • Probability of A given B: P(A|B) = ( \frac{P(A \cap B)}{P(B)} )
  • Similarly, P(B|A) = ( \frac{P(A \cap B)}{P(A)} )

Bayes' Theorem

• Formula Use:
  • Used to find probability of a person suffering an accident.
  • Accident Prone Example:
    • 30% accident prone, chance of accident is 40%.
    • Not accident prone chance is 20%.
    • Total Probability: P(E) using P(E|A) and P(A).

Conclusion and Next Steps

• Next class: Continue with conditional probability and introduce random variables.