Probability¶
Suggested Prerequisites¶
Overview¶
Probability Space¶
A probability space: \((\Omega, \mathcal{F}, P)\) where
\(\Omega\) is the sample space (set of all possible outcomes)
\(\sigma\)-algebra \(\mathcal{F}\) is a set of events where each event is an outcome
\(P\) is a function giving the probability of an event
Probability Function (\(P\))¶
Probability of an event must be greater than or equal to zero, probability of the sample space is equal to one, and probabilities of an intersection are additive:
\(P(A) \geq 0\)
\(P(\Omega) = 1\)
\(P(A_1 \cup A_1 \cup ...) = P(A_1) + P(A_2) + ... \) for any countable collection of mutually exclusive events \(A_1, A_2, ...\)
Consequences¶
\(P(o_i) = 1/n \) in a sample space \(\Omega = \{ o_1, o_2, ..., o_n\}\) where each outcome \(o_i\) is equally likely to occur \(\hspace{5pt} \forall \hspace{5pt} i = 1, ..., n\)
\(P(\bar{A}) = 1 - P(A)\) where \(\bar{A}\) is the complement of event \(A\)
For events \(A\) and \(B\), \(P(A\cup B) = P(A) + P(B) - P(A\cap B)\)
If \(A\subseteq B\), then \(P(A) \leq P(B)\)
For any event A, \(P(A) = \sum_{k=1}^m P(A \cap B_k)\) where mutually exclusive events \(B_1, B_2, ..., B_m\) such that \(B_1 \cup B_2 \cup ... \cup B_m\) and \(P(B_i) > 0 \hspace{5pt} \forall \hspace{5pt} i \)