By L.E. Sigler
ISBN-10: 0387901930
ISBN-13: 9780387901930
E-book through Sigler, L.E.
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Sample text
For the mathematical establishment, as represented by Bernoulli, the calculus of probability represented the perfect reasoning of a reasonable person in the absence of complete knowledge. For them, mathematics was a prescriptive way of enabling the rest of humanity to reach proper conclusions that they could not reach if left to their own devices. For d'Alembert, the central flaw in the calculus of probability was that it ought not to be a measure of ignorance concerning our understanding of a situation, but should instead be a way to describe the psychology of risk for the common person.
In 1718, she brought the detailed knowledge of the procedure back to England and eventually convinced Caroline, the Princess of Wales and future Queen of England, to have her two children inoculated, which she did in 1721. 5 What makes this deadly disease interesting in the present discussion is that a small percentage of those being inoculated contracted the disease and died; consequently becoming inoculated was a gamble. The odds in this life-and-death wager were one in seven that a person would contract smallpox at some time in their life without innoculation, versus a one in two hundred chance of dying from smallpox within a few months of being inoculated.
8 Probability Fig. 7: The vertcal axis is the information gained in bits by tossing a biased coin. The horizontal axis is the degree of bias measured by the probability of one outcome in a binary process. 36 • Where Medicine Went Wrong of a head coming up on a toss. In fact the figure is drawn for all possible weightings of the coin, making it either more or less likely to come up heads. It is evident from the symmetry of the information curve that the maximum information gain per toss is associated with a fair game, that is, one bit of information is obtained when the outcomes are equally likely and the probability of each outcome is one-half.
Exercises in Set Theory by L.E. Sigler
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