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This document begins by discussing the problem of determining the probability of an event occurring based on the number of times it has happened and the number of times it has failed to happen. The author introduces the work of Jacob Bernoulli, who posed the problem of determining the probability of an event occurring based on empirical observations. Thomas Bayes, on the other hand, is interested in determining the probability of an event occurring when all we know is the number of times it has happened and failed to happen. Bayes uses the example of a billiard table to illustrate his point. He rolls a ball across the table and records the number of times a second ball stops to the right of the first ball (success) and the number of times it stops to the left (failure). Bayes then deduces the probability of the location of the first ball based on these observations.

The primary application of Bayes’ system is in updating probabilities based on new information. The author explains that the Bayesian system allows for the comparison of posterior probabilities (updated probabilities) with the prior probabilities (original probabilities) based on new information. This approach recognizes that in a dynamic and uncertain world, there is no single answer.

The document then gives an example of a typical application of Bayesian analysis. In this example, a pin-manufacturing company has two factories, one older and one newer. The author poses the question of which factory is more likely to have produced a defective pin, given that a customer has complained about finding a defective pin. The prior probability suggests that the defective pin is most likely to have come from the newer factory, which produces 60% of the total output. However, when the priors are revised to reflect the additional information that the newer factory produces only one-third of the company’s total defective pins, the probability that the older factory is the culprit increases to 57.2%. This new estimate becomes the posterior probability.

The document concludes by emphasizing the boldness and audacity of Jacob Bernoulli, Abraham de Moivre, and Thomas Bayes in tackling the unknown and measuring uncertainty. The author highlights the progress made in the understanding of probability and risk in the 18th century, and anticipates further advances in the Victorian era.

Overall, the document provides a historical overview of the work of Jacob Bernoulli, Abraham de Moivre, and Thomas Bayes in the field of probability and risk analysis. It explores the problem of determining probabilities based on empirical observations and the use of Bayesian analysis to update probabilities based on new information. The document emphasizes the importance of measuring uncertainty and the boldness of the thinkers who contributed to the development of probability theory in the 18th century.

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