Week 11
Modelling Uncertainty: Probability

Uncertainty surrounds us. Probability gives us the tools to model it.

Probability is a much younger branch of mathematics than geometry or algebra which trace their roots to prehistory. In fact we know exactly when mathematicians first turned their eyes to probability, when a friend sought their advice about gambling! 🤑

A gambler's dispute in 1654 led to the creation of the mathematical theory of probability by two famous French mathematicians, Blaise Pascal and Pierre de Fermat. Antoine Gombaud, Chevalier de Méré, a French nobleman with an interest in gaming and gambling questions, called Pascal's attention to an apparent contradiction concerning a popular dice game. The game consisted in throwing a pair of dice 24 times; the problem was to decide whether or not to bet even money on the occurrence of at least one "double six" during the 24 throws. A seemingly well-established gambling rule led de Méré to believe that betting on a double six in 24 throws would be profitable, but his own calculations indicated just the opposite.
Source: http://homepages.wmich.edu/~mackey/Teaching/145/probHist.html

This week we'll learn about the tools of probability needed to answer the Chevalier de Méré's question and others. The main ones are both pictures rather than equations: tree diagrams, and Venn diagrams. Quite a change from the equation-heavy calculus we've been doing.

Before you begin an assignment or exercise

In working with you on exercise and assignment problems during office hours I see many of you flipping through notes, or windows on-screen, looking for formulas you need. Here's my tip for doing exercise and assignment problems: Begin by preparing a one-page summary of the week's material, and keep it at your side.

That way you'll have all the formulas and diagrams you need handy, and preparing the page is an excellent way of warming up to do the problems. Even better, when the time comes to create your cheat sheet for the final exam you can just make it up from your 13 weekly formula pages. Some weeks I have provided a formula sheet that you can use as a starting point (print it out and annotate it), but when I haven't I strongly encourage you to create one of your own. Cheatsheets rule! 😎

Little glitches

You'll notice when you get to it that the numbers in Problem 3 of the third exercise set don't add up. I've noted that in the exercise solutions. Also the titles of the solutions for the first two exercise sets refer to "homework", but they are solutions to the right problems! Sorry for the glitches, but I ran out of time to correct them. (And then there are the glitches I haven't even noticed yet...🤪).

Office Hours

As always I'll be in my Zoom "office" again this week on Monday, Wednesday and Friday from 10:30 am to 11:30 am. Come on by if you have questions.

That's all for now!

Enjoy the light and stay in touch,
Tim