Probability
Basic Concepts
- Experiment: An action with uncertain outcome
- Sample Space (S): Set of all possible outcomes
- Event (E): Subset of sample space
- Favorable Outcomes: Outcomes that satisfy the event
Probability Formula
- P(E) = Number of favorable outcomes / Total number of outcomes
- Range: 0 ≤ P(E) ≤ 1
- Impossible event: P(E) = 0
- Sure event: P(E) = 1
- P(not E) = 1 - P(E)
Types of Events
- Independent Events: P(A and B) = P(A) × P(B)
- Mutually Exclusive: P(A or B) = P(A) + P(B)
- Complementary Events: P(A) + P(not A) = 1
Important Formulas
- Addition Rule: P(A∪B) = P(A) + P(B) - P(A∩B)
- Multiplication Rule: P(A∩B) = P(A) × P(B|A)
- Conditional Probability: P(A|B) = P(A∩B) / P(B)
Common Examples
- Coin toss: P(Head) = 1/2, P(Tail) = 1/2
- Dice roll: P(any number) = 1/6
- Playing cards: Total = 52, Suits = 4, Each suit = 13
- Two dice: Total outcomes = 36
Playing Cards
- Total cards = 52
- Red cards (Hearts, Diamonds) = 26
- Black cards (Clubs, Spades) = 26
- Face cards (J, Q, K) = 12
- Number cards = 40
- Aces = 4
Quick Tips
- Always find total possible outcomes first
- For "at least one" problems, use P(E) = 1 - P(none)
- For independent events, multiply probabilities
- Probability is always between 0 and 1