2 Probability

For the following section, \(A\) and \(B\) represent events in the sample space \(S\).

2.1 Axioms

\(\mathbb{P}(A) \geq 0 \quad \forall A \subset S\)
\(\mathbb{P}(S) = 1\)
If \(A \cap B = \emptyset\), then \(\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B)\)

2.2 Union Rule

\(\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B)\)

2.3 Inclusion-Exclusion Principle

\(\mathbb{P}(A \cup B \cup C) = \mathbb{P}(A) + \mathbb{P}(B) + \mathbb{P}(C) - \mathbb{P}(A \cap B) - \mathbb{P}(A \cap B) - \mathbb{P}(B \cap C) + \mathbb{P}(A \cap B \cap C)\)

2.4 De Morgan’s Laws

\((A \cup B)^c = A^c \cap B^c\)

\((A \cap B)^c = A^c \cup B^c\)

2.5 Conditional Probability

\(\mathbb{P}(A|B) = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}\)

\(\mathbb{P}(A) = \mathbb{P}(A|B)\mathbb{P}(B) + \mathbb{P}(A|B^c)\mathbb{P}(B^c)\)

2.6 Bayes’ Theorem

\(\mathbb{P}(B_j|A) = \frac{\mathbb{P}(A|B_j)\mathbb{P}(B_j)}{\mathbb{P}(A)} = \frac{\mathbb{P}(A|B_j)\mathbb{P}(B_j)}{\sum_{i=1}^k \mathbb{P}(A|B_i)\mathbb{P}(B_i)}\)

2.7 Independence

If events \(A\) and \(B\) are independent: \(\mathbb{P}(A|B) = \mathbb{P}(A)\)

2.8 Counting Examples

There are \(n!\) ways to arrange \(n\) distinct elements in an ordered list.
There are \(6^n\) outcomes for \(n\) tosses of a \(6\)-sided die.