# 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 $$P(A \cup C | B) = P(A|B) + P(C|B)$$

• $$P(A \cap B) = P(A) * P(B|A)$$
• $$P(B) = P(A \cap B) + P(A^c \cap B)$$
• $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
• $$P(A \cap B) = P(B) * P(A|B) = P(A) * P(B|A)$$

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: $$P(C|D) \geq P(C|E)$$