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An incorrect model of the lottery, and when it doesn’t matter

Recently I came across an interesting idea about little mistakes in counting problems that actually don’t amount to much. In A Problem Squared 030, Matt Parker was investigating the question “What are the odds of having the same child twice?” and made some simplifying assumptions when thinking about DNA combinatorics. He justified leaving out a small number of things when counting an astronomical number of things by going through an example from the lottery.

The current UK lottery uses 59 balls and draws 6 of these, so the one in 45 million figure arises from \(\binom{59}{6}=45,\!057,\!474\), and the probability of winning is a tiny

\[ \frac{1}{45057474} = 0.00000002219387620 \text{.}\]

Matt posits the idea that somewhere along the way we forget to include some tickets.

But let’s say along the way while I’m working it out, for strange reasons I go ‘oh you know what, I’m going to ignore all the options which are all square numbers. You know, I just can’t be bothered including them. Yeah, they’re legitimate lottery tickets, but just to make the maths easier I’m going to ignore them’. And people are getting up in arms, and they’re like ‘you can’t ignore them, they’re real options’.

The Aperiodical is 10!

Not that we’re overly consumed with numerical coincidences, but it’s perhaps nice to note that ten years ago today we made a little fuss of launching a new blog site with our first post, a post marking Felix Klein’s 163rd birthday, and a video about the Klein Bottle featuring Matt Parker and Katie Steckles. space containing Aperiodical-related items to explore. Visible is a big Aperiodical logo as well as logos for the Carnival of Mathematics, the Mathematical Objects podcast and The Big Internet MathOff.
Our 10th birthday party space in

Mathematical Objects: Hairy ball

Mathematical Objects

A conversation about mathematics inspired by a hairy ball. Presented by Katie Steckles and Peter Rowlett.

Hairy ball

Adapting my ‘programming for mathematicians’ module for teaching during the COVID-19 pandemic

When teaching moved online due to COVID-19, we had to quickly work out how to deliver our modules online. The main options used to replace in-person classes were:

  • pre-recorded videos followed by live online tutorials for students to get support while completing exercises;
  • live online classes offering a mixture of lecturer delivery and student activity.

The first option is good for a module with lots of content delivery, such as when learning new mathematical techniques. In modules with some content delivery but a focus on interaction and discussion, such as mathematical modelling, the second is a good choice.

I felt neither was quite right for my second-year programming module. I opted instead for delivering notes and exercises which students could work through when convenient (which might be in a designed class time or might not) and used my time on the module to write responses to student queries and give feedback on programs written as formative work.

In class students tend to say they’ve done an exercise correctly and because you’re walking round a computer room it can be hard to examine their code in detail. Spending time looking at what they submit as ‘correct’ code in greater detail, it became clear that often there are subtle issues which can be usefully discussed in considered feedback.

Overall, I think this semi-asynchronous delivery was much better use of time and I was able to view more code and give better feedback than I would in-person.

I wrote about my experience delivering this module through the pandemic – the end of one academic year and the whole of the next – with Alex Corner in an open-access article which has just been published as ‘Flexible, student-centred remote learning for programming skills development‘.

This is part of a special issue of International Journal of Mathematical Education in Science and TechnologyTakeaways from teaching through a global pandemic – practical examples of lasting value in tertiary mathematics education. There are loads of articles with useful reflections and good ideas that emerged from pandemic teaching.

If you are interested in pandemic literature in higher education teaching and learning, I’m aware of two other journal special issues you might like:

Mathematical Objects: Superegg with Hannah Fry

Mathematical Objects

A conversation about mathematics inspired by a superegg. Presented by Katie Steckles and Peter Rowlett, with special guest Hannah Fry.

Superegg, a type of oval-like solid

British Science Week mathematicians poster competition

Maria Gaetana Agnesi by Bianca Milesi Mojon (1836) and 祖冲之铜像.jpg by 三猎.

I wrote a mathematics-themed competition for British Science Week, which is a UK-wide event lasting ten days taking place this month.

The competition calls for individuals or groups to research the life and/or work of a mathematician and produce a poster to share their findings. The six mathematicians available to choose from are: