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

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 y y0 for all values x 2 [0; 0:99 \Delta x0 ] is "more probable" to work well than a design for which y y0 only for the values x 2 [0; 0:5 \Delta x0 ]. In this paper, we describe an interval computationsrelated 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