By William Feller

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1. Of hundred such simulations, on average 5 would fail to satisfy the bound. In the more interesting case when the probability p is small, 100/p ﬂips are required approximately in order to have a “reliable” estimate of the unknown value of p . ¯ can also be used to estimate general probabilities p = As shown next, X P(A) of a statement A about an outcome of an experiment that can be performed inﬁnitely many times in an independent manner. 3 Note that the value of p is also unknown. 5 0 20 40 60 Number of flips Fig.

Ak be a partition of S , and B1 , . . , Bn , . . a sequence of true statements (evidences). If the statements B are conditionally independent of Ai then the a posteriori odds after receiving the n th evidence qin = P(Bn | Ai )qin−1 , n = 1, 2, . . , where qi0 are the a priori odds. e. replaced by the posterior odds. This recursive estimation of the odds for Ai is correct only if the evidences B1 , B2 , . . are conditionally (given Ai is true) independent. In the following example, presenting an application actually studied with Bayesian techniques by von Mises in the 1940s [54], we apply the recursive 28 2 Probabilities in Risk Analysis Bayes’ formula to update the odds.

This will be discussed in the last chapter. 2 Non-stationary streams Computations are often done for stationary situations; however, most real phenomenon are non-stationary: simple environmental conditions vary with 44 2 Probabilities in Risk Analysis time, new safety technologies or regulations are introduced, systems deteroriate with time. Since introducing non-stationarity complicates mathematical modelling of uncertainties one often neglects it. However, there are situations when the non-stationary character of a problem is essential when safety of a system is evaluated.