What are the chances of a class with 24 students having 3 pairs of students sharing a birthday?

Yesterday we looked at the famous Birthday problem – how many people do you need to have in a room to have a 50/50 chance of two people having the same birthday? That project is here:

Diving into the Birthday problem with kids

Today we continued the project (with just my older son as my younger son was hiking) and studied the problem that originally motivated this project -> If you have 24 students in a class, what is the chance that exactly 3 pairs of students will share a birthday? This is the surprisingly fun situation in my son’s English class.

We will – as I think it standard for the introductory version of this problem – be making the assumption that all birthdays are equally likely. If you want to see a really neat discussion – though not really a math for kids paper! – see the paper in this tweet:

So, to start the project today we first reviewed the main ideas from yesterday:

Next we took a step towards solving the problem by looking at the chance of having exactly 2 pairs. Once piece of the counting here is tricky, so we used the computer to help see what the problem was.

Now we tackled the “exactly 3 pairs problem”:

Finally, I had my son make up a problem to solve – he decided to find the chance of all 24 students pairing up. This problem wasn’t too hard given the prior work. It was also a fun challenge to try to estimate the chance of this happening.

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