1.5 Conditional Probability 1. Conditional Probability ●●●●● 2. The multiplication rule 3. Partition Theorem 4. Bayes rule
1.5 Conditional Probability 1. Conditional Probability 2. The multiplication rule 3. Partition Theorem 4. Bayes’ Rule
●●● ●●●● ●●●●● ●●●● ●●●●● 1.5. 1 Conditional Probability ●●●● Conditioning is another of the fundamental tools of probability probably the most fundamental tool. It is especially helpful for calculating the probabilities of intersections, such as P(AnB Additionally, the whole field of stochastic processesl"机过程 is based on the idea of conditional probability. What happens next in a process depends, or is conditional, on what has happened beforehand
1.5.1 Conditional Probability ⚫ Conditioning is another of the fundamental tools of probability: probably the most fundamental tool. It is especially helpful for calculating the probabilities of intersections, such as P(A∩B). ⚫ Additionally, the whole field of stochastic processes随机过程 is based on the idea of conditional probability. What happens next in a process depends, or is conditional, on what has happened beforehand
●●● ●●●● ●●●●● ●●●● ●●●●● Dependent events ●●●● ●●●● Suppose a and b are two events on the same sample space There will often be dependence between a and b This means that if we know that b has occurred. it changes our knowledge of the chance that a will occur
Dependent events ⚫ Suppose A and B are two events on the same sample space. There will often be dependence between A and B. This means that if we know that B has occurred, it changes our knowledge of the chance that A will occur
●●● ●●●● ●●●●● ●●●● ●●●●● Example: Toss a dice once ●●●● Let event A=“geta6” Let event B=‘“ get an even number” If the die is fair, then P(A)=I andP(B)=2 However if we know that B has occurred. then there is an increased chance that a has occurred P(A occurs given that B has occurred)=3.F result 6 esult 2 or 4 or We write P(A given B)=P(A B)=3 Question: what would be P(B A)? P(B A)=P(B occurs, given that A has occurred P( get an even number, given that we know we got a 6)
Example: Toss a dice once
●●● ●●●● ●●●●● ●●●● Conditioning as reducing the sample space ●●●●● ●●●● We throw two dice. Given that the sum of the eyes is 10. what is the probability that one 6 is cast? Let b be the event that the sum is 10 B={(4,6),(5,5),(6,4)} et a be the event that one 6 is cast. {(1,6),…,(5,6),(6,1)…,(6.5)} Then, AnB=((4, 6), (6, 4)). And, since all elementary events are equally likely, we have (|B)22/36P(A∩B) 33/36P(B) This is our definition of conditional probability
Conditioning as reducing the sample space ( ) ( ) 3/ 36 2 / 36 3 2 ( | ) P B P A B P A B = = =
●●● ●●●● ●●●●● ●●●● ●●●●● 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
●●● Simple example using Conditional ●●●● ●●●●● ●●●● Probabilities ●●●●● ●●●● We toss a fair coin three successive times. We wish to find the conditional probability P(A B)when A and B are the events A=more heads than tails come up), B=(lst toss is a head y The sample space consists of eight sequences, Q=HHH, HHT, HTH, HTT, THH, THT, TTH, TTTH which we assume to be equally likely. The event B consists of the four elements HHH, HHT, HTH, HTT, so its probability is 4 P(B) The event An B consists of the three elements outcomes HHH, HHT, HTH, so its probability is P(A∩B) Thus, the conditional probability P(AJB)is P(A/B)=P(AnB)3/8_3
Simple Example using Conditional Probabilities
●●● ●●●● ●●●●● 1.5.2 The Multiplication Rule ●●●● ●●●●● ●●●● ●●●● Assuming that all of the conditioning events have positive probability, we have P4)=P4p(4(An)P=4) The above formula can be verified by writing n24)=P(4) 4∩42)P(41∩A2∩42) P(4)P(41∩A2) A For the case of just two events, the multiplication rule is simply the definition of conditional probability P(4n4)=P4P(44
1.5.2 The Multiplication Rule
