1. **Stating the problem:** You want to learn about classical probability, starting from the basics and moving to more advanced concepts. 2. **Definition of classical probability:** The classical definition applies when an experiment has $n$ equally likely outcomes. The probability $P(E)$ of an event $E$ is given by $$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{m}{n}$$ where $m$ is how many outcomes favor event $E$. 3. **Key properties:** - Probability values satisfy $0 \leq P(E) \leq 1$. - The sum of probabilities of all possible mutually exclusive, collectively exhaustive outcomes equals 1: $$\sum_{i=1}^{n} P(E_i) = 1$$ 4. **Mutually exclusive events:** Two events are mutually exclusive if they cannot happen at the same time. If $E_1$ and $E_2$ are mutually exclusive, then $$P(E_1 \cap E_2) = 0$$ This means no overlap in outcomes. 5. **Collectively exhaustive events:** A set of events is collectively exhaustive if at least one event must occur when the experiment happens. The sum of their probabilities is 1. 6. **Example: Rolling a fair six-sided die** - Total outcomes: $n=6$ (one each for 1 to 6 spots). - All outcomes are equally likely with probability $\frac{1}{6}$. Classical probability that the die shows a 4-spot: $$P(4) = \frac{1}{6}$$ 7. **Higher level concepts:** - Probability of complementary event $E^c$ is $$P(E^c) = 1 - P(E)$$ - Probability of union of mutually exclusive events $E_1, E_2$ is $$P(E_1 \cup E_2) = P(E_1) + P(E_2)$$ - For independent events (extending classical concepts), $$P(E_1 \cap E_2) = P(E_1) \times P(E_2)$$ 8. **Using the dice example for compound events:** Probability of rolling an even number (2, 4, or 6): Number of favorable outcomes $=3$ $$P(\text{even}) = \frac{3}{6} = \frac{1}{2}$$ 9. **Summary:** Classical probability is a simple ratio counting favorable to total outcomes assuming equality of likelihood. It applies well to dice, cards, coins, and any fair experiment. This introduces both foundational and some extended probability concepts systematically.