Dirty Pages of Logarithm Tables, Lifetime of the Universe, and (Subjective) Probabilities on Finite and Infinite Intervals

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In many engineering problems, we want a physical characteristic y to lie within given range Y; e.g., for all possible values of the load x from 0 to x0, the resulting stress y of a mechanical structure should not exceed a given value y0. If no such design is possible, then, from the purely mathematical viewpoint, all possible designs are equally bad. Intuitively, however, a design for which yy0 for all values x ∈ [0,0.99 ⋅ x0] is “more probable” to work well than a design for which yy0 only for the values x ∈ [0,0.5 ⋅ x0]. In this paper, we describe an interval computations-related formalization for this subjective notion of probability. We show that this description is in good accordance with the empirical distribution of numerical data and with the problems related to estimating the lifetime of the Universe.

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