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Unformatted text preview: Introduction to Probability COS 341 Fall 2004 Basic Laws of Probability Definition 1 A sample space S is a nonempty set whose elements are called outcomes . The events are subsets of S . Since events are subsets, we can apply the usual set operations to events to obtain new events. For events A and B , the event A B represents the set of outcomes that are in both event A and event B , i.e. A B represents the event A and B . Similarly, A B represents the event A or B . Definition 2 A probability space consists of a sample space S and a probability function Pr () , mapping the events of S to real numbers in [0 , 1] , such that: 1. Pr ( S ) = 1 , and 2. If A , A 1 , . . . is a sequence of disjoint events, then Pr [ i A i ! = X i Pr ( A i ) . (Sum Rule) One consequence of this definition is the following: Pr ( A ) = 1- Pr ( A ) . (Complement Rule) Several basic rules of probability parallel facts about cardinalities of finite sets: Pr ( B- A ) = Pr ( B )- Pr ( A B ) (Difference Rule) Pr ( A B ) = Pr ( A ) + Pr ( B )- Pr ( A B ) (Inclusion-Exclusion) An immediate consequence of (Inclusion-Exclusion) is Pr ( A B ) Pr ( A ) + Pr ( B ) (Booles Inequality) Similarly (Difference Rule) impies that If A B , then Pr ( A ) Pr ( B ). (Monotonicity) Example 1 Suppose we wire up a circuit containing a total of n connections. The probability of getting any one connection wrong is p . What can we say about the probability of wiring the circuit correctly ? (The circuit is wired correctly iff all the n connections are made correctly.) 1 solution: Let A i denote the event that connection i is made correctly . So Pr ( A i ) = p . Pr (all connections correct) = Pr ( n i =1 A i ) . Without any additional assumptions (on the dependence of the events A i ), we cannot get an exact answer. However, we can give reasonable upper and lower bounds. Pr ( n i =1 A i ) Pr ( A i ) = 1- p Pr ( n i =1 A i ) = 1- Pr ( n i =1 A i ) = 1- Pr ( n i =1 A i ) 1- n X i =1 Pr ( A i ) = 1- np Both these bounds are tight, i.e. we can construct situations where the correct answer is equal to the upper bound and those where the correct answer is equal to the lower bound. Conditional Probability Definition 3 Pr ( A | B ) denotes the probability of event A given that event B has occured....
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