EXAMPLE: What is the analog for uncountable sample spaces of theequally likely case for discrete sample spaces? Suppose 0,cforsomec0. WritePabfor the probability of event:ab. Thenassumethe probability isPabb−acfor any 0≤ab≤c. This is the length of the intervala,bover thelength of the entire interval for the sample space.50
∙In this example, all particular values have zero probability ofoccurring, that is,Pa0 for all possible outcomes 0≤a≤c.Why? Supposeais strictly between 0 andc; the endpoints are handledsimilarly. Then for integersjlarge enough so thata−1/j0 anda1/jc,Pa−1/ja1/j2/cj. ButPa≤Pa−1/ja1/jbecausea⊂a−1/j,a1/j.So forall jsufficiently large,Pa≤2/cj, which can only happen ifPa0.51
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∙Uncountable spaces are intuitively tricky: any particular outcome haszero probability (yet, when we turn to statistics and collecting data, weeventually see an outcome). In such cases, we focus on the probabilityof events such as intervals.∙For the “equally likely” model, the probability of events involvingintervals of the same length is the same – regardless of where theinterval starts. For example, ifc100,P020P8010020/1001/5.∙Later, we use the notion of continuous random variables to representuncountable sample spaces.52