Overview

This topic explores the mathematical tools used to count possibilities and evaluate probabilities. You’ll learn how to calculate outcomes using multiplication, combinations, and permutations, and apply probability rules to real-world situations involving chance and uncertainty.

Key Concepts and Structures

• Multiplication Rule: If one event can occur in m ways and another in n ways, there are m × n total outcomes.
• Factorials: n! = n × (n-1) × (n-2)... × 1. Used in permutations and combinations.
• Permutations: Arrangements where order matters.
  Formula: P(n, r) = n! / (n - r)!
• Combinations: Selections where order does not matter.
  Formula: C(n, r) = n! / [r!(n - r)!]
• Probability: Measures the chance of an event.
  Formula: P(event) = favorable outcomes / total outcomes
• Complement Rule: P(not A) = 1 - P(A)
• Union and Intersection:
  • P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
  • P(A ∩ B) = probability both occur
• Independent Events: One event does not affect the other. P(A ∩ B) = P(A) × P(B)
• Mutually Exclusive Events: Cannot happen at the same time. P(A ∩ B) = 0
• Expected Value: The predicted average outcome.
  Formula: E = Σ [value × probability]

Step-by-Step Example

Problem: A coin is flipped and a die is rolled. What is the probability of getting heads and a 4?

Step 1: Identify outcomes:
- Coin: 2 outcomes (heads, tails)
- Die: 6 outcomes (1–6)

Step 2: Total outcomes = 2 × 6 = 12

Step 3: Favorable outcome = heads and 4 → only 1 possibility

Final Answer: 1/12

Quick Tip