How Many Rolls?

I roll a fair 6-sided die until I get my first 6. How many rolls on average does that take?

We start by rolling lots of dice, plotting the data and using these to get an empirical estimate of that theoretical average. Then we use the power of recursive thinking to find the theoretical answer which turns out to be 6. This is an example of a more general result that we call the dartboard theorem.

Of the 40 trials:

4 got the first 6 on the 1st roll,

6 got it on the 2nd roll,

3 got it on the 3rd roll,

etc. 

This activity is a real stretch for grade 9, but we want them to see and participate in a grown-up analysis. In addition, we want them to do some thinking that will extend their good work on probability from Grade 8. Now is the time.

This rich problem will open the way to a bunch of even more-sophisticated recursive arguments in Grade 10 with How many rolls II.