Ordinary language reasoning about probabilities. It is this middle level that is the main subject of this book. It may avoid numbers entirely, as in “proof beyond reasonable doubt” in law or the nonnumerical judgments of plausibility that scientists and detectives make in evaluating their hypotheses. Or it may involve rough numerical estimates of probabilities, as in racecourse odds and guesses about the risks of rare events. 3. Formal mathematical reasoning of the kind found in textbooks of probability and statistics. The higher levels may be more noble and perfect, but they are so at a cost:
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