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 = = =