Let be the set of all such s the set of events can be

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Unformatted text preview: + x + x + ··· = 1−x 2 ∞ xk = or k=0 1 . 1−x (1.4) The formula (1.3) for partial sums and the formula (1.4) for infinite sums of a geometric series are used frequently in these notes. 3 “ω ” is a lower case omega. 1.2. AXIOMS OF PROBABILITY 11 Example 1.2.4 (Repeated binary trials) Suppose we would like to represent an infinite sequence of binary observations, where each observation is a zero or one with equal probability. For example, the experiment could consist of repeatedly flipping a fair coin, and recording a one each time it shows heads and a zero each time it shows tails. Then an outcome ω would be an infinite sequence, ω = (ω1 , ω2 , · · · ), such that for each i ≥ 1, ωi ∈ {0, 1}. Let Ω be the set of all such ω ’s. The set of events can be taken to be large enough so that any set that can be defined in terms of only finitely many of the observations is an event. In particular, for any binary sequence (b1 , · · · , bn ) of some finite length n, the set {ω ∈ Ω : ωi = bi for 1 ≤ i ≤ n} should be in...
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