The Birthday “Paradox”

I asked the kids what they’d like to do for a math project today and my younger said that he’d like to do something birthday related. Well, that’s easy 🙂

In the first part of the project I introduced them to the famous Birthday Paradox and asked them what they thought the answer was. The question is easy to state:

How many people do you need to have in a room so that you have a 50% chance of 2 people sharing the same birthday.

Their initial guess for the number of people was 180 (younger son) and 240 (older son):

After the short introduction we went on to talk about how to calculate the probability. My younger son had the nice idea to start from some simple cases of 1, 2, and 3 people in the room. Eventually we got to the idea of looking at the problem via complimentary counting.

Now that we had some experience with the complimentary counting approach, we did one more example – this time with 4 people. The kids noticed that the probabilities are increasing pretty quickly.

Finally we went to my work computer to look at the probabilities for various cases using Excel. The kids were really surprised to see how few people were required to solve the problem:

So, a fun project because the result is so non-intuitive. Also a neat project to use with kids because there are lots of basic math ideas we use to solve the problem – percents, decimals, and counting techniques to name a few. Nice little birthday morning!