1. Probability Space

A Probability Space is a mathematical framework for modeling random experiments. It consists of three components:

1. Sample Space (S)

   • The set of all possible outcomes of an experiment.
   • Example: Tossing a coin → $S = \{H, T\}$

2. Event Space ($\mathcal{F}$)

   • A collection of events, which are subsets of the sample space.
   • Example: Event "Head occurs" → $E = \{H\}$

3. Probability Measure (P)

   • A function that assigns a probability to each event in $\mathcal{F}$.
   • Must satisfy axioms of probability.

2. Axioms of Probability (Kolmogorov’s Axioms)

Let $E$ be any event in a sample space $S$. Then:

1. Non-negativity

   $$P(E) \ge 0$$
   • Probability can’t be negative.

2. Normalization

   $$P(S) = 1$$
   • The probability of the sample space is 1 (certainty).

3. Additivity (for mutually exclusive events)

   $$\text{If } E_1 \cap E_2 = \emptyset, \text{ then } P(E_1 \cup E_2) = P(E_1) + P(E_2)$$
   • Probability of union of disjoint events is sum of their probabilities.

3. Basic Laws/Rules of Probability

1. Complementary Rule

• $E'$ = event that does not occur

2. Addition Rule

• For any two events $A$ and $B$: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
• If $A$ and $B$ are mutually exclusive, then $P(A \cap B) = 0$, so: $$P(A \cup B) = P(A) + P(B)$$

3. Multiplication Rule

• For independent events $A$ and $B$:

  $$P(A \cap B) = P(A) \cdot P(B)$$

• For dependent events:

  $$P(A \cap B) = P(A) \cdot P(B|A)$$

• Where $P(B|A)$ = conditional probability of $B$ given $A$ occurs.

4. Total Probability Law

• If events $E_1, E_2, ..., E_n$ form a partition of the sample space: $$P(A) = \sum_{i=1}^{n} P(A|E_i) \cdot P(E_i)$$

5. Bayes’ Theorem (Inverse Probability)

• Used to update probabilities based on new evidence.

4. Summary

Key Points to Remember

• Probability spaces provide a formal foundation for probability theory.
• Laws of probability ensure probabilities are consistent and can be used for computations.
• Understanding dependent vs independent events is crucial for multiplication rule.