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The renowned mathematical physicist Pierre-Simon, Marquis de Laplace wrote in his opus on probability in 1812 that “the most important questions of life are, for the most part, really only problems in probability”. His words ring particularly true today in this the century of “big data”.
This introductory course takes us through the development of a modern, axiomatic theory of probability. But, unusually for a technical subject, the material is presented in its lush and glorious historical context, the mathematical theory buttressed and made vivid by rich and beautiful applications drawn from the world around us. The student will see surprises in election-day counting of ballots, a historical wager the sun will rise tomorrow, the folly of gambling, the sad news about lethal genes, the curiously persistent illusion of the hot hand in sports, the unreasonable efficacy of polls and its implications to medical testing, and a host of other beguiling settings. A curious individual taking this as a stand-alone course will emerge with a nuanced understanding of the chance processes that surround us and an appreciation of the colourful history and traditions of the subject. And for the student who wishes to study the subject further, this course provides a sound mathematical foundation for courses at the advanced undergraduate or graduate levels.
The course is divided into five topical segments which taken together constitute a self-contained introduction to mathematical probability. Read on for a bird's-eye view of these topics.
Topic I: Towards an axiomatic theory of chance
We begin with the stirrings of a mathematical theory in the 17th century in the resolution of a historical wager of the Chevalier de Méré and follow the developments in understanding leading to the modern axiomatic foundations of probability established in the 20th century.
Topic II: From side information to conditional probabilities
In this segment we shall encounter unanticipated challenges to intuition when presented with side information about a probability experiment and discover the subtle importance of additivity in a tongue-in-cheek exhortation on the survival of our species.
Topic III: Independence—the warp and the woof in the fabric of chance
The distinctive and rich intuitive content of the theory of probability and its link to observations in physical experiments is provided by the notion of statistical independence. We follow the progress from multiplication tables to a formal notion of independence, with enticing applications in a casino game, genetics, and sports psychology to whet the appetite.
Topic IV: From polls to bombs and queues—enter the binomial and the Poisson
We next turn to ruminations on the implausible efficacy of small polls in tracking sentiments of large populations and discover how the hugely important binomial distribution emerges from these considerations. We then retrace the historical discovery of a curious approximation to the binomial and stumble upon the fascinating Poisson distribution. We promptly explore applications ranging from the merely weighty to the diverting and macabre, the distribution of bomb hits in London in World War II being particularly memorable.
Topic V: The fabulous limit laws—the bell curve pirouettes into the picture
The law of large numbers is a cornerstone of probability responsible for much of its intuitive content and it is fitting that our gradual development concludes with it. Among its rich applications, we discover why polls work and the implications to medical testing. The remarkable bell curve now takes centre stage completing the triad of fundamental distributions and we discover why polls really work.