By Luis Garrido
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Laplace first discussed the probability of causes, apparently unaware of Bayes’s work, in a paper of 1774. Using Bernoulli’s model, he showed that if from an infinitely large urn the first ( p + q) draws had yielded p black and q white balls, the probability of a further black ball on the ( p + q + 1)th draw, assuming that all possible fractions of black balls in the urn were initially equally likely, is ( p + 1)/( p + q + 2). So if every one of m drawn balls is black, the probability is (m + 1)/(m + 2) that the next one will be so too.
The value of money, he argued, is not constant for a rational man. Unless you are particularly greedy or profligate, an extra dollar means less if you are rich than if you are poor. Illustrating the point with examples from the world of commerce, Bernoulli suggested that the value of money went as the logarithm of the amount held, and was thus able to produce a non-infinite entry fee to play the St. 15 As a model of rational judgment, therefore, 14 15 Jorland 1987 reviews attempts to solve the St.
Thus did the Rule of Succession justify induction. In his paper of 1774, Laplace used his method to solve the problem of points, before moving to more practical matters and showing that for three measurements at least, the mean was a better average than the median. In 1778 he wrote on the binomial case of male-to-female birth ratios, and subsequently worked from distributions of samples to estimates of population sizes. He was also able to invert a famous problem in astronomy and find the probability that some cause, rather than chance, had resulted in orbits of the planets being so nearly coplanar.