📐 1. Basic Counting Principles

Counting is the foundation of probability. Understanding how many ways an event can happen is crucial before assigning probabilities.

🔡 1.1 Counting Arrangements ("Words")

"Words" = ordered combinations of symbols/letters (not necessarily actual words)

➤ With Repetition Allowed

• Problem: How many words of length k can be made from an alphabet of n letters?
• Solution: Each letter has n choices → total = n × n × ... × n = nᵏ

➤ Without Repetition (Permutations)

• Case 1: Using all k letters (alphabet of size k)

  → All k letters used once, total permutations = k!

• Case 2: Using k letters from a larger alphabet of size n (k < n)

  → We select and arrange k letters from n

  → Total = n! / (n - k)!

  (Ordered selections of k elements from n)

🧮 1.2 Counting Subsets (Combinations)

• Unordered selections of size k from n elements

  → Total = C(n, k) = n! / (k! (n - k)!)

• Key idea: Start with permutations, then divide by the number of ways to reorder k elements (i.e., k!)

• Example: Choosing 3 students from a group of 5 → C(5, 3) = 10

🎯 2. Introduction to Probability

Probability quantifies how likely an event is to happen, assuming a well-defined sample space of equally likely outcomes.

📊 2.1 Formal Definition

If an outcome O can occur in f favorable ways out of n total possible outcomes:

P(O) = f / n

• Value range: 0 ≤ P(O) ≤ 1