| Basic Relationships of Probability |
|---|
For a given event A, the complement of A is the event
consisting of all outcomes that are not in A. The complement of A
is denoted by Ac or just A'.
In any probability application, either event A or its complement
A
\[
\Pr [A] + \Pr \left[ A^c \right] = 1 \qquad \Longrightarrow \qquad
\Pr [A] = 1 - \Pr \left[ A^c \right] .
\]
Theorem: (inclusion-exclusion)
- For any two events A and B:
\[ \Pr \left[ A \cup B \right] = \Pr \left[ A \right] + \Pr \left[ B \right] - \Pr \left[ A \cap B \right] . \]
- For any three events A, B, and C:
\[ \Pr \left[ A \cup B \cup C \right] = \Pr \left[ A \right] + \Pr \left[ B \right] + \Pr \left[ C \right] - \Pr \left[ A \cap B \right] - \Pr \left[ A \cap C \right] - \Pr \left[ B \cap C \right] + \Pr \left[ A \cap B \cap C \right] . \]
- For arbitrary number of events:
\[ \Pr \left[ \omega_1 \cup \omega_2 \cup \cdots \cup \omega_n \right] = \sum_{i=1}^n \Pr \left[ \omega_i \right] - \sum_{i\ne j} \Pr \left[ \omega_i \cap \omega_j \right] + \cdots + (-1)^{n+1} \Pr \left[ \omega_1 \cap \omega_2 \cap \cdots \cap \omega_n \right] . \]
Theorem: (monotonicity) If for two events we have A⊆B, then \( \Pr \left( A \right) \le \Pr \left( B \right) . \)