We are working through Mosteller’s 50 Challenging Problems in Probability and today’s problem was a nice surprise -> the famous coupon collector problem.

Here’s how I introduced it and what the boys thought the answer would be:

Next the boys talked through how to think about the problem. The diagram they ended up drawing was such a great surprise! At the end of this video the boys were a little stuck on the problem of trying to figure out how many steps it would take, on average, to go from having one coupon to having two.

Now we dive into the calculation of the number of steps required to go from one coupon to having two coupons. This is a tricky calculation but we broke it into pieces and had a good guess at the answer by the end of this video:

Now that we had a good guess at the answer for the sum of the complicated series, my older son found a pretty clever way to evaluate the sum. That was a really nice break through!

By the end of this video we were wondering about how to find the expected number of steps it would take to go from 2 coupons to three coupons:

The next step was trying to sum the series that would give us the expected number of steps that we would need to go from 2 steps to 3 steps. Once we made that calculation the boys saw how to evaluate the remaining terms.

Finally, we went to a little matahematica program to check our results and to see what the maximum number of trials would look like.

This was a super project. I loved hearing the ideas that the boys had and was super proud that they were able to work through all the way to the solution of the original problem.