Probability

# Probability - Chapter 1 Probability Frank Porter February 1...

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Unformatted text preview: Chapter 1 Probability Frank Porter February 1, 2011 1.1 Definition of Probability The notion of probability concerns the measure (“size”) of sets in a space. The space may be called a Sample Space , or Event Space . We define a probability as a set function according to: Definition 1.1 Let S be a (sample) space. Let P ( E ) be a real additive set function defined on sets E in S ; P is called a probability function if the following conditions are satisfied: 1. If E is a subset ( event ) in S , then P ( E ) ≥ . 2. P ( S ) = 1 . 3. If E,F ∈ S and E ∩ F = ∅ , then P ( E ∪ F ) = P ( E ) + P ( F ) . If S is an infinite sample space, and E 1 ,E 2 ,E 3 ,... is an infinite sequence of disjoint events in S , then P ( ∪ ∞ i =1 E i ) = ∞ X i =1 P ( E i ) . (1.1) In the language of measure thory, this definition is equivalent to: Definition 1.2 Let S be a (sample) space. A probability P defined on S is a measure on S such that: P ( S ) = 1 . (1.2) The motivation for this definition is the idea that we may have a set of possible outcomes, with each outcome having some likelihood of occurring, given by its probability measure. The sample space contains all possible outcomes, and the measure is normalized such that the probability over the whole space is unity. 1 2 CHAPTER 1. PROBABILITY Note that we shall use the terms outcome , sampling , and event interchangeably in the present context. We ignore here the issue of whether all possible subsets of S may be assigned a probability measure; the sets we shall be concerned with are measurable. For example, consider S 3 ≡ { a,b,c } . The measure generated by P ( a ) = P ( b ) = P ( c ) = 1 / 3 defines a probability on S 3 . In particular, note that P ( S 3 ) = P ( a ) + P ( b ) + P ( c ) = 1, according to the additive property of a measure. Probability typically is used as the mathematical tool dealing with “ran- domness” or “uncertainty”. The concept of randomness is represented by the process of sampling an outcome from a set of possible outcomes, where the ac- tual outcome is unknown until the sampling has occurred. In the above example with S 3 , the elements a,b,c of the sample space may be given a concrete real- ization in the form of three distinct balls to be drawn blindly from a container. We don’t know prior to the drawing which ball we will get. We only know that the probability of getting a specified ball is 1 / 3. The concept of uncertainty permits an extension of the application of proba- bility beyond describing random processes. For example, it may be that in some trial process, only one outcome will actually occur. That is, the sample space in reality has only one element. However, we may be ignorant, prior to the trial, concerning the element. In this case, a larger sample space may be defined, with probabilities assigned to describe our uncertainty in what the true element is....
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Probability - Chapter 1 Probability Frank Porter February 1...

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