Chapter: Probability

• Introduction

  • Random Experiment
  • Outcome
  • Sample Space
  • Real-life applications (Weather, Games)

• Empirical (Experimental) Probability

  • Based on actual observations/experiments
  • Formula: $P(E) = \frac{\text{Number of trials in which event happened}}{\text{Total number of trials}}$
  • Limitation: Results vary with number of trials

• Theoretical (Classical) Probability

  • Based on reasoning and equally likely outcomes
  • Formula: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
  • Notation: $P(E)$ represents probability of event $E$

• Fundamental Terms & Rules

  • Elementary Event: An event having only one outcome
  • Sum of Probabilities: $P(E) + P(\text{not } E) = 1$
  • Complementary Event: $\bar{E}$ or $P(\text{not } E) = 1 - P(E)$
  • Sure Event (Certain Event): Probability = $1$
  • Impossible Event: Probability = $0$
  • Range of Probability: $0 \le P(E) \le 1$

• Standard Problems & Examples

  • Coin Toss Problems
    • Single coin (Head, Tail)
    • Two coins ($HH, HT, TH, TT$)
    • Three coins
  • Die/Dice Problems
    • Single die ($1, 2, 3, 4, 5, 6$)
    • Even/Odd numbers
    • Prime numbers
    • Numbers greater/less than a specific digit
  • Playing Cards (Deck of 52)
    • Total cards: $52$
    • Red cards ($26$): Hearts, Diamonds
    • Black cards ($26$): Spades, Clubs
    • Face Cards ($12$): Kings, Queens, Jacks
    • Honor/Aces cards
  • Bag/Urn Problems
    • Balls of different colors (Red, Blue, White)
    • Drawing marbles or discs at random