Main Formulas
Some Axioms
- \(P(A) + P(A^c) = 1\)
- \(P(A \cup B) = P(A) + P(B) - P(A \cap B) = P(A) + P(A^c \cap B)\)
- \(P(A \cup B) \leqslant P(A) + P(B)\) - union bound
- \(P(A \cup B \cup C) = P(A) + P(A^c \cap B) + P(A^c \cap B^c \cap C)\)
From De Morgan’s laws
- \((S \cap T)^c = S^c \cup T^c\)
- \((S \cup T)^c = S^c \cap T^c\)
- \(P(A \cap B) = 1 - P(A \cap B)^c = 1 - P(A)^c \cup P(B)^c = 1 - (1 - P(A)) \cup (1 - P(B))\)
Conditional Probability (probability of A given B)
- \(P(A|B) = \frac{P(A \cap B)}{P(B)}\) if \(P(B) > 0\)
- \(P(A|B) \geqslant 0\) assuming \(P(B) > 0\)
- \(P(\Omega | B) = \frac{P(\Omega \cap B)}{P(B)} = 1\)
- if \(A \cap C = \varnothing\), then \(P(A \cup C | B) = P(A|B) + P(C|B)\)
Radar example:
- \(P(A \cap B) = P(A) * P(B|A)\)
- \(P(B) = P(A \cap B) + P(A^c \cap B)\)
- \(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
- \(P(A \cap B) = P(B) * P(A|B) = P(A) * P(B|A)\)
Total probability theorem:
\begin{equation*}
P(B) = \sum_{i}P(A_i) * P(B|A_i)
\end{equation*}
Bayes’ rule
\begin{equation*}
P(A_i | B) = \frac{P(A_i) * P(B | A_i)}{\sum_j P(A_j) * P(B | A_j)}
\end{equation*}
\begin{equation*}
P(A|B) = \frac{P(A) * P(B|A)}{P(B)}
\end{equation*}
Independence
- \(P({A, B, C}) = P(A) + P(B) + P(C)\)
- \(P(A \cap B) = P(A) * P(B)\)
- \(P(A \cap B) = P(A)P(B|A) = P(B)P(A|B)\)
- \(P(A|B) = P(A)\)
- \(P(A \cap B^c) = P(A) P(B^c)\)
- \(P(A^c \cap B^c) = P(A^c) P(B^c)\)
Conditional independence
\(P(A \cap B | C) = P(A|C) * P(B |C)\)
Reliability
\(P(A \cap B) = P(A) * P(B)\)
Serial: \(P(a \rightarrow b) = p^k\)
Parallel: \(P(a \rightarrow b) = 1 - (1 - p)^k\)
Conditional probability of smaller subset
Given events \(C, D, E\) such that:
- \(D \subset E\)
- \(C \cap D = C \cap E\)
than: \(P(C|D) \geq P(C|E)\)