# Decisions under uncertainty : probabilistic analysis for by Ian Jordaan

By Ian Jordaan

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

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If all the n events are incompatible, there are n constituents. The numbers of constituents, say k, is therefore such that n ≤ k ≤ 2n . To summarize, we have shown that any set of events {E 1 , E 2 , . . , E n } can be expressed in terms of a partition of constituents C1 , C2 , . . , Ck , where k is in the range just noted. The probabilities of the constituents must normalize, as before. 2 Venn diagrams; probability of compound events Events can be illustrated by means of Venn diagrams. These assist considerably in visualizing and understanding relationships using logical notation.

Then A ∧ B = AB is the event that both the Titanic and the iceberg are in the degree-rectangle. ) If A = 1 and B = 1 then AB = 1 (both logically and arithmetically). Similarly if A or B or both are equal to 0, then AB = 0, as required. We shall use either AB or A ∧ B for the logical product. The logical sum A ∨ B means ‘A or B or both’. In terms of the example above, either the Titanic or the iceberg, or both, are in the degreerectangle. The notation A + B for the logical sum is not consistent with the notation A ∨ B, which is an event with values 0 or 1 depending on its falsity or truth.

Or we are designing a new space shuttle and do not know the probability of success. The difference from the ‘first limit’ above is that we do not now have a ‘well-defined’ probability, and we wish to use the results of experiments to infer the probabilities. The problem then is one of inference. 4). We would expect that in the first limit noted, repeated trials of the same experiment would ‘in the long run’ give us relative frequency ratios that are close to our ‘well-defined’ values. We would of course become suspicious on obtaining the 100th, 1000th, .