How should we interpret chance around us? Watch beautiful mathematical ideas emerge in a glorious historical tapestry as we discover key concepts in probability, perhaps as they might first have been unearthed, and illustrate their sway with vibrant applications taken from history and the world around.
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.
- Santosh Venkatesh - Electrical and Systems Engineering