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# ch03 - 3 AXIOMS OF PROBABILITY 11 3 Axioms of Probability...

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3 AXIOMS OF PROBABILITY 11 3 Axioms of Probability The question here is: how can we mathematically define a random experiment? What we have are outcomes (which tell you exactly what happens), events (sets comprising certain outcomes), and probability (which attaches to every event the likelihood that it happens). We need to agree on which properties these objects must have in order to compute with them and develop a theory. When we have finitely many equally likely outcomes all is pretty clear and we have already seen many examples. However, as is common in mathematics, infinite sets are much harder to deal with. For example, we will see soon what it means to choose a random point within a unit circle. On the other hand, we will also see that there is no way to choose at random a positive integer — remember that “at random” means all choices are equally likely, unless otherwise specified. A probability space is then a triple (Ω , F , P ). The first object Ω is an arbitrary set of outcomes, sometimes called a sample space . The second object F is the collection of all events, that is a set of subsets of Ω. Therefore, A ∈ F necessarily means that A Ω. Can we just say that each A Ω is an event? In this course, you can assume so without worry, although there are good reasons for not assuming so in general ! I will give the definition of what properties F needs to satisfy, but this is just for illustration and you should take a course in measure theory to understand what is really going on. Namely, F needs to be a σ -algebra , which means (1) ∅ ∈ F , (2) A ∈ F = A c ∈ F , and (3) A 1 , A 2 , · · · ∈ F = ⇒ ∪ i =1 A i ∈ F . What is important is that you can take complement A c of an event A (i.e., A c happens when A does not happen), unions of two or more events (i.e., A 1 A 2 happens when either A 1 or A 2 happens), and intersection of two or more events (i.e., A 1 A 2 happens when both A 1 and A 2 happen). We call events A 1 , A 2 , . . . pairwise disjoint if A i A j = if i negationslash = j — that is, at most one of such events can occur. Finally, the probability P is a number attached to every event A and satisfies the following three axioms: Axiom 1 . For every event A , P ( A ) 0. Axiom 2 . P (Ω) = 1. Axiom 3 . If A 1 , A 2 , . . . is a sequence of pairwise disjoint events, then P ( uniondisplay i =1 A i ) = summationdisplay i =1 P ( A i ) . Whenever we have an abstract definition like this, the first thing to do is to look for examples. Here are some.

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3 AXIOMS OF PROBABILITY 12 Example 3.1. Ω = { 1 , 2 , 3 , 4 , 5 , 6 } , P ( A ) = (no. of elements in A ) 6 . The random experiment here is rolling a fair die. Clearly this can be generalized to any finite set with equally likely outcomes. Example 3.2. Ω = { 1 , 2 , . . . } , and you have numbers p 1 , p 2 , . . . 0 with p 1 + p 2 + . . . = 1. For any A Ω, P ( A ) = summationdisplay i A p i .
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ch03 - 3 AXIOMS OF PROBABILITY 11 3 Axioms of Probability...

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