Lecture Notes: Probability and Complementary Events

Introduction

• Clarification of the term "complementary events" in math.
  • Not about receiving compliments.
  • Definition: Complementary events are opposite outcomes.
    • Example: If you eat 2 slices from an 8-slice pizza, the complementary event is leaving 6 slices.
    • Another Example: Watching TV for 4 hours means not watching for 20 hours.

Probability Basics

• Formula for Probability:
  • Probability = ( \frac{\text{Number of events being watched}}{\text{Number of possible events}} )

Example Problem with a Deck of Cards

Part A: Probability of Drawing an Ace

• Deck Information: 52 cards total.
  • 4 aces (one per suit: clubs, spades, diamonds, hearts).
• Calculation:
  • ( \frac{4}{52} = \frac{1}{13} )
  • Result: The probability of drawing an ace is ( \frac{1}{13} ).

Part B: Probability of Not Drawing an Ace (Complementary Event)

• Concept: Total probability equals 1.
  • Either draw an ace or do not draw an ace.
• Calculation:
  • Probability of not drawing an ace = 1 - Probability of drawing an ace.
  • ( 1 - \frac{1}{13} = \frac{12}{13} )
  • Result: The probability of not drawing an ace is ( \frac{12}{13} ).

Practice Problem: Blocks in a Bag

Part A: Probability of Drawing a Red Block

• Bag Information: 12 blocks total.
  • 3 red blocks.
• Calculation:
  • ( \frac{3}{12} = \frac{1}{4} )
  • Result: The probability of drawing a red block is ( \frac{1}{4} ).

Part B: Probability of Not Drawing a Red Block (Complementary Event)

• Calculation:
  • Probability of not drawing a red block = 1 - Probability of drawing a red block.
  • ( 1 - \frac{1}{4} = \frac{3}{4} )
  • Result: The probability of not drawing a red block is ( \frac{3}{4} ).

Key Takeaways

• Complementary events represent opposite outcomes.
• Total probability of all possible outcomes adds up to 1.
• For probability of something not happening, subtract the probability of it happening from 1.