## Probability

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 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.

• What resources will I need for this class?

You will need a broadband internet connection to access the video lectures, an index finger to hit "pause" frequently while you absorb the material, and pencil and paper to write down the mathematics to help with the absorption.

• What is the coolest thing I'll learn if I take this class?

Hmm.  A case could be made for any of the examples listed in the course description.  But I'll select the unreasonable efficacy of polls, partly because the mathematics is "cool" and the historical narrative captivating, but also because of the ubiquitous impact of polls in day to day life.  By rights a small poll capturing the opinions of, say, 1200 people has no business saying anything meaningful at all about the inclinations of a large population of, say, one billion individuals. But, unbelievably, it does (if done properly). This, the concluding lecture in the series, knits together the entire theory and produces a potent and powerful application as an end product: the cottage industry that we call polls. This is the reason why medical testing works, why marketing companies spend small fortunes on polling the sentiments of small groups, and how modern politicians keep a finger on the pulse of the electorate.

• Who should take this class?

In the modern age there is scarce an area of endeavour left untouched by probabilistic and chance-driven considerations and I would make the case that any educated individual in the 21st century should have a basic understanding of chance phenomena. To be sure, a mathematical understanding of chance processes requires some basic mathematical training and background. In our case this translates into exposure to calculus and accordingly this class nominally targets high school students and beginning undergraduates with the appropriate background. But more broadly any individual with the requisite background in basic mathematics and a curiosity about how chance processes shape the world will find this class interesting and perhaps even compelling.

### Recommended Background

With all its historical context and diverse modern applications, this is fundamentally a class on mathematical probability.  As background a student should have a solid exposure to at least one semester of calculus (as is typically covered in a freshman college class or in the AP or IB programmes).  A student should be comfortable with algebraic and functional notation for variables, sets, and functions. She should be familiar with the ideas of sequences and limits and have seen the common convergent series.  Exposure to differential and integral calculus is not essential for this course.

In addition to basic calculus, we will need some elementary ideas from combinatorics and the language of sets.  I've provided background review lectures on these topics. These are made available with the preview.

Now, to be sure, all these things I've listed are necessary background elements but above all the most important things a student should bring with her are a liking for mathematical reasoning, a devouring curiosity about how things work, and a delight in intellectual discovery.