1. 🔍 Formalizing Probability

Probability becomes truly rigorous when built on the concept of a probability space, which links outcomes to numerical likelihoods in a mathematically consistent way.

2. 🎯 Sample Space (𝑆)

The sample space is the set of all possible outcomes of an experiment.

• Denoted by S
• The outcomes can be:
  • Discrete (e.g., dice rolls: {1, 2, 3, 4, 5, 6})
  • Structured (e.g., grid positions on a chessboard)
  • Complex (e.g., outcomes of a horse race or a lottery)
• Not all outcomes need to be equally likely!

🧠 Examples:

• Rolling a die → S = {1, 2, 3, 4, 5, 6}
• Rolling two dice and summing → S = {2, 3, ..., 12}
• Horse race outcomes → S = {Black Caviar, Kingston Town, Farlap, ...}
• Lottery → S = {Grand prize, Second prize, ..., No prize}

3. 📏 Probability Measure (𝑃)

A probability measure assigns a number p(s) to each outcome s in S, satisfying:

1. Non-negativity: 0 ≤ p(s) ≤ 1
2. Normalization: Σ p(s) = 1 (over all s ∈ S)

🔍 Examples:

• Dice sum probabilities:
  • P(2) = 1/36, P(7) = 6/36, etc.
• Horse race:
  • Probabilities may be assigned based on historical data or betting odds.

4. 🧩 Probability Space (𝑆, 𝑃)