How worried should I be about the positive result on my blood test?
Is the beetle with the rare markings actually a rare beetle?
Probability gives us the tools to model uncertainty.
Probability is a much younger branch of mathematics than geometry or algebra both of which can trace
their roots to prehistory. In fact we know exactly when mathematicians first turned their
eyes to probability: when a friend sought 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
Visual Models
This week we'll learn about the tools of probability needed to answer the Chevalier de Méré's
question, and others. There are some (simple) formulas, but the main tools are both pictures rather than equations: tree diagrams,
and Venn diagrams. Quite a change from the equation-heavy calculus we've been doing. 😅
Monday. "Simple" probability, which is enough to answer the Chevalier's question.
Wednesday. Conditional probability and Bayes' theorem which will let us decide about that medical test and the beetle.
Friday. How to use Venn diagrams to organize our data so we can calculate probabilities.
Advice: Before you begin an assignment or exercise
Pro 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.
Preparing the page is
an excellent way of warming up to do the problems, and then you'll have all the formulas and diagrams you need handy. 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 (by printing it out and annotating it), but when I haven't I strongly encourage you to create
one of your own. Cheatsheets rule! 😎
Collaborate! Do research! Just don't copy.
A reminder about how I feel about students working together, and about students doing research (google) to find answers: I love both these things!
Working with other people can help you catch small errors you could get stuck on, suggest
approaches that would not have occurred to you, and keep your spirits up. If you're stuck on
a problem, e.g. a proof, turning to the web is a natural thing to do. The goal is for you to
learn and come to an understanding of the course material. If other people or other resources
can help you do that then I'm all for them. Of course you shouldn't just copy someone else's
work, because that wouldn't help you form an understanding. And you should acknowledge your
sources when you've looked things up. But you already know that. 😏
Watch out! 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...🫣).
