Probability: From Basics to Advanced

Basic Formula

P(Event) = (Favorable outcomes) / (Total outcomes)

Key Rules

• 0 ≤ P(E) ≤ 1
• P(not E) = 1 - P(E)
• P(A or B) = P(A) + P(B) - P(A and B)
• P(A and B) = P(A) × P(B) [if independent]

Conditional Probability

• P(A|B) = P(A and B) / P(B)
• Bayes' Theorem: P(A|B) = [P(B|A) × P(A)] / P(B)

Common Scenarios

• Dice: Total outcomes = 6ⁿ (n dice)
• Cards: 52 cards, 4 suits, 13 ranks
• Coins: Total outcomes = 2ⁿ (n coins)

Expected Value

E(X) = Σ [x × P(x)]

CAT Patterns

• At least one: Use complement (1 - none)
• Exactly k: Use combinations
• Conditional: Use tree diagrams

Practice Tip

Draw probability trees for complex multi-stage problems.