Chapter Summary: Probability

This chapter introduces the fundamental concepts of probability, dealing with uncertainty, chance, and the prediction of outcomes in random experiments.

1. Fundamental Concepts

• Probability: A measure of uncertainty regarding the occurrence of an event. It quantifies the chance of an event happening.
• Experiment: An action that results in well-defined outcomes.
• Random Experiment: An experiment where:
  • There is more than one possible outcome.
  • All possible outcomes are known in advance.
  • The specific outcome cannot be predicted beforehand.
• Trial: A single performance of an experiment (e.g., one toss of a coin or one throw of a die).
• Event: Each outcome of an experiment is called an event.
• Equally Likely Outcomes: When all outcomes of an experiment have an equal chance of occurring (e.g., a fair coin or a fair die).

2. Calculation of Probability

3. Important Properties of Probability

• Range: The probability of an event always lies between 0 and 1 (inclusive).
  Mathematically: 0 ≤ P(E) ≤ 1
• Sure Event: An event that is certain to happen has a probability of 1.
  Example: Getting a natural number less than 7 when throwing a standard die.
• Impossible Event: An event that can never happen has a probability of 0.
  Example: Getting a number greater than 6 when throwing a standard die.
• Sum of Probabilities: The sum of the probabilities of all possible outcomes of an experiment is always 1.
• Validity: Probability can never be negative and can never be greater than 1. Answers like -2 or 1.25 are impossible.

4. Common Sample Spaces (Outcomes)

Tossing Coins

• 1 Coin: Total outcomes = 2 (Head, Tail).
• 2 Coins (or 1 coin tossed twice): Total outcomes = 4 (HH, HT, TH, TT).
• 3 Coins: Total outcomes = 8 (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT).

Throwing Dice

• 1 Die: Total outcomes = 6 (1, 2, 3, 4, 5, 6).
  • Even numbers: 2, 4, 6
  • Odd numbers: 1, 3, 5
  • Prime numbers: 2, 3, 5
• 2 Dice: Total outcomes = 36 (Pairs ranging from (1,1) to (6,6)).

Playing Cards

• A standard deck contains 52 cards.
• Calculations are based on suits (Hearts, Diamonds, Spades, Clubs), colors (Red, Black), or face cards (King, Queen, Jack).