Statistics: The Birthday Paradox

This has always been one of those problems that I have loved exploring. I recently found a website that will model the outcome in a brilliantly beautiful way. Here’s a little more about it from the Birthday Problem Wikipedia page:

In probability theory, the birthday problem or birthday paradox concerns the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday.

By the pigeonhole principle, the probability reaches 100% when the number of people reaches 367 (since there are only 366 possible birthdays, including February 29). However, 99.9% probability is reached with just 70 people, and 50% probability with 23 people. These conclusions are based on the assumption that each day of the year (excluding February 29) is equally probable for a birthday.

It may seem surprising that the probability is above 50% when there is a pair with the same birthday for a group as small as 23 individuals. This is made more plausible when considering that a comparison will actually be made between every possible pair of people rather than fixing one individual and comparing him or her solely to the rest of the group.

https://en.wikipedia.org/wiki/Birthday_problem

The Pudding’s Russell Goldenberg has created this website to demonstrate this paradox.

Click to visit The Birthday Paradox Experiment website

Have a visit, have a play and contribute to the data.

I would absolutely use this as a hook the next time I need to tackle statistics with an upper Key Stage 2 class!

Linkpudding.cool/2018/04/birthday-paradox/

Opportunities to notice – developing number sense and reasoning

I have long been a fan of Steve Wyborney’s blog where he shares plenty of engaging and different ideas to develop mathematical thinking. Although based in Oregon, the ideas are still very relevant to the UK curriculum.

I recently came across his use of dot patterns to promote ‘noticing’ – not something we necessarily spend a lot of time on, but I now wonder why we don’t.

One big focus in mathematics teaching currently is the use of small step and space to let children think. This sort of activity undoubtedly lends itself towards these ideas.

Steve has created 10 pages of dotted patterns, the purpose of which is to promote a wide variety of partitioning, seeking different perspectives and recording observations in ways that can be very useful. I would guarantee there will be children in every class who see a different pattern or way of breaking things up to draw out different mathematics every time.

He recommends using 18 copies of the same pattern with the space underneath being vital to allow plenty of working room and many opportunities to explore the same thing in different ways.

The following image is from his explanation video of how to use the patterns (found here), demonstrating 4 ways to investigate one such pattern.

For a greater challenge, once children are used to working with these, you could develop more complex images with different coloured dots – each colour being worth a separate value – to explore larger numbers, perhaps. Each dot doesn’t need to necessarily only be worth 1, either.

The 10 patterns and some further information can be found at the link below.

Link: 180 Opportunities to Notice

Prime Number GIF

This lovely animated GIF makes me feel like there is something worth getting children to investigate further – either as a lesson activity or as a project to create short films showing other numbers.

As always, I’d love to hear if you use the idea, or it has inspired you to do something different in a lesson.