Week 11
Modelling Uncertainty: Combinatorics

Business

The start of the week email. No need to reread it if you received it. It's just linked here in case anyone accidentally deleted it (it's happened before 🥺).

⦿ Monday. Essential combinatorics

For us combinatorics means three things, one principle and two formulas: the Multiplication Principle, and the formulas for Permutations and Combinations.
The multiplication principle Pencast 9:06| PDF
For two independent events A and B that can occur in m and n ways respectively, the total number of ways A⋂B can occur is m×n.
Password combinatorics Pencast 11:19| PDF
Why you should use long passwords drawn from large sets of characters, and four-digit PIN codes are an abomination.
Lineups and partial lineups: Permutations Pencast 12:02| PDF
Permutations count the number of ways r items chosen from n items can be ordered.
Committees and subsets: Combinations Pencast 7:22| PDF
Combinations count the number of subsets of r things that can be selected from n things.
Combinatorics: Formula sheet PDF
The week's formulas summarized.
Example: Whitehorse vs "the communities" Pencast 10:09 | PDF
Is Whitehorse dominating committee memberships? What are the chances that it's a coincidence?
Example: Playing the lottery Pencast 14:04| PDF
What are our chances of winning at Lotto Max?
Example: Eating pizza Pencast 14:31| PDF
How many Dominos pizzas are possible?

Exercises

If you're comfortable with the topics above you're ready to work on:

Supplementary resources:

§3.5 in Chapter 3 of College Mathematics for Everyday life by Inigo, Jameson, Kozak, Lanzetta, and Sonier covers combinatorics.

⦿ Wednesday. Distributions of outcomes

Quincunx? Screencast 4:39
The ubiquitous bell curve/Gaussian distributon/normal curve (when you're ubiquitous you get named multiple times!) can arise from the sum of many individual random events. You can play with the Quincunx simulation yourself, or watch a video of an actual physical quincunx.
Quincunx! Pencast 16:14| PDF
Calculating Quincunx outcomes manually using Pascal's Triangle, and formulaically using combinations.
Back to coin flipping, and probability Pencast 13:32| PDF
Calculating probabilities of many, many tree paths, e.g. the probability of getting 7 heads in 10 tosses of a coin. This turns out to be a useful model for many, many phenomena.
Bernoulli trials and binomial distributions Pencast 10:16| PDF
If the Quincunx and coin flipping are the same, then of course mathematicians formalized and named their commonalities.

Exercises

If you're comfortable with the topics above you're ready to work on:

Supplementary resources:

The Math is Fun folks who provide the simulator have a good discussion of the Quincunx.

In College Mathematics for Everyday life by Inigo, Jameson, Kozak, Lanzetta, and Sonier
§4.3 covers the binomial distribution.

⦿ Friday. Case study: Mark-recapture

Mark and recapture Pencast 21:36| PDF
We can rarely go and count all the individuals in an animal population. Instead we have to take a sample and infer the population from the sample. Mark-recapture is a common way of doing this.

Exercises

If you're comfortable with the topics above you're ready to work on:

Supplementary resources:

I used the simplest possible mark-recapture model and formula here. Given its centrality to wildlife population management extensive work has been done on mark-recapture strategies and modelling. If you want to know more about Mark-Recapture techniques Chapter 2 Estimating Abundance and Density: Mark-Recapture Techniques in Ecological Methodology 3ed by C.J. Krebs is one well-written resource that is available online.

Assignment

Assignment 10 is posted. Due by 1 pm Friday March 22. Note that the problems are worth different numbers of marks. Use your time wisely.

Name the file you submit A10_firstname.pdf, e.g.

A10_Tim.pdf

If you have to submit multiple files, add a suffix number to each, e.g.

A10_Tim_1.pdf
A10_Tim_2.pdf