Set Theory & Probability Axioms

1. Sets and Events

• Union ($\cup$): $A \cup B$ means event A happens OR event B happens.
• Intersection ($\cap$): $A \cap B$ means event A AND event B happen simultaneously.
• Mutually Exclusive: If $A \cap B = \emptyset$, the events cannot happen together.

2. Counting Methods

• Permutations: Order matters. $P(n, r) = \frac{n!}{(n-r)!}$
• Combinations: Order does not matter. $C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$

3. Kolmogorov's Axioms

1. The probability of any event is non-negative: $P(E) \ge 0$.
2. The probability of the entire sample space is 1: $P(\Omega) = 1$.
3. For mutually exclusive events, $P(A \cup B) = P(A) + P(B)$.

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