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Basic Principle of Counting

Probability helps us determine how likely an event is to happen. To put simply, probability is counting the number of ways an event can happen divided by the number of total possible outcomes that can happen.

\[ \text{Probability of Event A} = \frac{\text{Number of outcomes where A occurs}}{\text{Total number of possible outcomes}} \]

The Basic Principle of Counting

If you are doing two things one after the other, and the first thing can happen in m ways, and for each of those, the second thing can happen in n ways, then the total number of possible outcomes is m × n.

Why this is true:

You can list out every possible combination. For each of the m options from the first thing, there are n options from the second thing. So you end up with m rows, and each row has n items. This means there are m × n total combinations.

Simple Examples

Example 1: Flipping Two Coins

Question: What is the probability of getting at least one heads?

  • Sample space \((Ω)\): \({\{HH, HT, TH, TT\}}\) → 4 outcomes
  • Event A: at least one heads → {HH, HT, TH} → 3 outcomes → \(P(A) = \frac{3}{4} = 0.75\)

Question: What is the probability of no heads?

  • Event B: no heads → \({TT}\) → 1 outcome → \((B) = \frac{1}{4} = 0.25\)

Example 2: Rolling Two Dice

Question: What is the probability that at least one die is a 5?

  • Total possible outcomes: 6 × 6 = 36
  • Favorable outcomes:
    • First die is 5 → (5,1) to (5,6): 6 outcomes
    • Second die is 5 but first is not → (1,5), (2,5), (3,5), (4,5), (6,5): 5 outcomes
    • Total favorable = 6 + 5 = 11
    • Probability: \(P(A) = \frac{11}{36}\)

Example 3: Choosing Samples

Question: There are 10 women. Each woman has 3 children. You want to pick one woman and one of her children for best mother-child pair award.

  • First, you have 10 choices for the woman (m options).
  • Then, for each woman, you have 3 choices (her children) (n options).
  • So the total number of different choices is: 10 × 3 = 30 choices.

Counting as the Foundation

When events become more complex (e.g., flipping 1000 coins, or rolling 15 dice), brute-force enumeration becomes impractical. This is where probability theory evolves into using:

  • Combinatorics (permutations and combinations)
  • Formulas and models (e.g., binomial theorem, distributions)
  • Set theory and logic (to define events clearly)

Key Assumptions (in Basic Models)

  1. Uniform Probability: All outcomes are equally likely (fair coin, fair die).
  2. Independence: Events do not affect each other (each flip or roll is independent).

What Is Random?

A process is “random” if we cannot predict its outcome with the information available.

  • Deterministic but unknown: A coin flip is physically deterministic, but too complex to model precisely—so we treat it as random.
  • Truly stochastic: Radioactive decay is genuinely random (quantum uncertainty).

Applications of Counting & Probability

  • Games (poker, dice, darts)
  • Weather forecasts
  • Medical diagnostics
  • Failure modeling in engineering
  • Financial risk assessment
  • Machine learning and AI