Lotteries, bookmaking and ancient randomizers: Local and global analyses of chance

A local analysis of a chance system recovers the properties of the totality by accumulating the chance properties of its components. A global analysis considers only the properties of the chance system taken as a whole. It can sometimes provide an easier demonstration of the fitness for purpose of chance systems. Such is the case for the operators of lotteries and for bookmakers at a racetrack. Physical randomizers akin to dice were used extensively in antiquity for gambling, drawing lots and in divination. There is no ancient account of the fitness of these randomizers for these purposes and no ancient theory of the individual chances of outcomes. A global analysis can establish their fitness for purpose. If the rules of gambling and lot drawing are such that everyone has an equal opportunity, they are fair, independently of the individual, local chances. The randomizers are fit for the purposes of divination in so far as it is believed that the oracle has no control on the outcomes of the randomizers. In asking why some earlier culture did not discover probability theory, we presume incorrectly an inevitability to probability theory. The better question is why any culture found the theory at all.