Week 12
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

Combinatorics for us means three things, one principle and two formulas: the Multiplication Principle, Permutations, and Combinations.
The multiplication principle 📹 Pencast 9:06| 📘 PDF
For two independent events A and B that can occur in mand 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 communities 📹 Pencast 10:09 📘 PDF
Is Whitehorse dominating committee memberships? What are the chances it's 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 phnomena.
Bernoulli trials and binomial distributions 📹 Pencast 10:16| 📘 PDF
If the Quincunx and coin flipping are the same, then of course matheamticians 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 one 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 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 available online.

All the exercises in one place

All due by 4 pm Monday March 29.

Exercise Set 11 Part 1: Combinatorics Problems and Solutions.

Exercise Set 11 Part 2: Binomial probabilities Problems and Solutions.

Exercise Set 11 Part 3: Mark-recapture likelihood via Excel Problems. The problem statement includes an image of the solution you can use to check your work.

Assignment

Assignment 10 is posted. Due by 4 pm Thursday April 1. Note that the problems are worth different numbers of marks. Use your time wisely.

Submit PDF files of work done by hand, or in a word processor. (Let me know if you are unsure how to create these.) Email is preferred, but you can also submit your work through Moodle.
Excel or other binary files should be sent as file attachments.
Google sheets can be submitted by sharing a link to them.
Work done in Desmos can be submitted by sharing a link to your saved graph.

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