2 - Review#
2.1 - Sample Space#
The sample space (\(\Omega\)) is the set of all possible outcomes of an experiment.
For example, the event that exactly two coins show heads when three are flipped is:
\[ A = \{ HHT, HTH, THH \} \]
2.2 - Formulae to Compute Probabilities of Events#
The Additive Law of Probability:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
If \(A\) and \(B\) are disjoint, then
\[ P(A \cup B) = P(A) + P(B) \]
2.3 - Conditional Probability#
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
where \(P(B) > 0\).
We say that \(A\) and \(B\) are pairwise independent if
\[ P(A) = P(A|B) \Longleftrightarrow P(B) = P(B|A) \Longleftrightarrow P(A \cap B) = P(A)P(B) \]