Conditional Probabilities

Recap: Conditional Frequencies

• When considering conditional frequencies, additional information can change the relative frequencies
  • So can the probabilities for the occurrence of events change, when additional information is received. \(\rightarrow\) Conditional probabilities

General Formula:

• If \(B \subseteq \Omega\) is an event with \(P(B > 0\), then the conditional probability of the event \(A \subseteq \Omega\) given (the condition) B is given by: \[P(A|B) = \frac{|A \cap B|}{|B|}\]

Alternative calculation methods for the conditional probability:

• \(P(B) = \frac{|B|}{|\Omega|} \rightarrow |B| = P(B) \cdot |\Omega|\)
• \(P(A|B) = \frac{|A \cap B|}{|B|} \rightarrow |A \cap B| = P(A \cap B) \cdot |Omega|\)

Calculation Rules

The calculation rules for probabilities carry over to conditional probabilities

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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Rule 6: Product Rule

• \(P(A \cap B) = P(A|B) \cdot P(B) \: \: \: \forall A,B, \subseteq \Omega\) with \(P(B) > 0\)

Rule 7: Law of total probability

• If \(\Omega = B_{1} \dot{\cup}B_2\dot{\cup}\dots \dot{\cup} B_k\) is a partition of the sample space into disjoint events, then for e very event \(A \subseteq \Omega\):
  • \(P(A) = \sum_{i = 1}^{k} P(A|B_i) \cdot P(B_i)\)

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Independence

Two events A and B are stochastically independent, if P(A and B) = P(A) * P(B)

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