●●● ●●●● ●●●●● ●●●● ●●●●● Definition of conditional probabili ●●●● Definition: Let A and B be two events. The conditional probability that event A occurs, given that event B has occurred, is written P(A B), and is given by P(A B P(A∩B) P(B) Conditional probability provides us with a way to reason about the outcome of an experiment based on partial information o Note: Follow the reasoning above carefully. It is important to understand why he conditional probability is the probability of the intersection within the new sample space Conditioning on event B means changing the sample space to B Think of P(A B)as the chance of getting an A, from the set of B's only
Definition of conditional probability Conditional probability provides us with a way to reason about the outcome of an experiment based on partial information
●●● Conditional Probabilities Satisfy the ●●●●● Three axioms ●●●● · Nonnegative: AB)≥0 · Normalization P(2∩B)P(B P[Ω2B P(B P( Additivity: If A, and A, are two disjoint events P(41U4B)=P (A1UA2)∩B) P(B p distributive P(A1∩B)U(A2∩B) P(B P(41∩B)+P(42∩B) disjoint sets P(B) =P(A4|B)+P(42|B)
Conditional Probabilities Satisfy the Three Axioms
●●● ●●●● ●●●●● Conditional Probabilities Satisfy General ●●●● ●●●●● ●●●● Probability laws Properties probability laws P(A, UA2 B)<P(AB)+P(42 B) P(41UA2|B)=P(1|)+P(4B)-P(41∩42|B) Conditional probabilities can also be viewed as a probability law on a new universe b because all of the conditional probability is concentrated on B
Conditional Probabilities Satisfy General Probability Laws