Excerpt from The Probability That a Numerical, Analysis Problem Is Difficult

The main justification for using a uniform distribution is that it appears to be fair: each problem is as likely as any other. However, it does not appear to apply in many practical cases for a variety of reasons, including the fact that any set of problems which can be represented in a computer is necessarily discrete rather than continuous. We will discuss the validity of our choice of uniform distribution as well as alternatives at length in section 6 below.

Finally, given this distribution, we must compute the induced probability distribution of the condition number. It turns out that all the problems we consider here have a common geometric structure which lets us compute the distributions of their condition numbers with a single analysis, which goes as follows.

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