Conditional Probability Rules

Rule 1: Conditional probabilities are numbers between 0 and 1

• \(0 \le P(A|B) \le 1 \: \: \: \forall A,B \subseteq \Omega \text{ with } P(B) > 0\)

Rule 2: The impossible event \(\varnothing\) has conditional probability \(0\)

• \(P(\varnothing|B) = 0 \:\:\: \forall B \subseteq |Omega \text{ with } P(B) > 0\)

Rule 3: If \(A \subseteq C\) holds, then event \(C\) has at least the same conditional probability as event \(A\)

• \(\text{For all } B \subseteq \Omega \text{ with } P(B) > 0 \text{ it holds that } P(A|B) \le P(C|B) \text{ if }A \subseteq C\)

Rule 4: Calculation fo the conditional probability of the complementary event

• \(P(\bar{A}|B) = 1 - P(A|B) \text{ for all } B \subseteq \Omega \text{ with } P(B) > 0\)

Rule 5: Addition rule (to calculate the conditional probability of the union)

• \(P(A \cup C|B) = P(A|B) + P(C|B) - P(A \cap C | B) \text{ for all } B \subseteq |Omega \text{ with } P(B) > 0\)

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