By Ian Jordaan

1. Uncertainty and decision-making -- 2. the concept that of likelihood -- three. Distributions and expectation -- four. the idea that of software -- five. video games and optimization -- 6. Entropy -- 7. Mathematical features -- eight. Exchangeability and inference -- nine. Extremes -- 10. danger, security and reliability -- eleven. facts and simulation -- 12. end

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17). 12), we can write an expression for the logical sum in terms of the products E 1 , E 1 E 2 , E 1 E 2 E 3 , and so on. The results all derive from coherence. 3 Set theory Set theory was developed for a broad class of situations – sometimes broader than the events or quantities of interest in probability theory. The sets of integers, of rational numbers, of real numbers, are all of interest in our work; sets of higher cardinality are not of relevance here. But sets can be used in probability theory: we have always a set of possible outcomes in our subject of study.

36 The concept of probability be used with set theory by means of an ‘indicator function’. For example, if X is the set of real numbers, and x is a particular value, then the indicator function I (x) could be defined such that I (x) = 1 if X ≤ x, and I (x) = 0 if X > x. Then I (x) ≡ E if E is the event X ≤ x in Boolean notation. The relation between Boolean notation and an indicator function of sets is illustrated as follows: Boolean notation Indicator function on sets A ∧ B or AB A∨B A∧ B ∨C I (A ∩ B) I (A ∪ B) I (A ∩ B ∪ C) We shall use the Boolean notation without an indicator function in the remainder of this book.

1) and Chapter 8. We have dealt with two limiting cases of ‘repetitions of the same experiment’. 3 The Titanic tosses of a die shows that the trials are not the same. The point is that we cannot distinguish them. What is common to each set of experiments whether we consider the probability ‘well defined’ or not? The simplest interpretation is that the order of the results is not important. In other words if we permute the set of results, for example, convert {heads, tails, heads} to {heads, heads, tails}, and judge these two sets of results to be equivalent, to have the same probability of occurrence, then the order is not important.