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 having 40 trials, each time rolling a die until we get a 6, and noting the number of rolls that it took. The data are plotted below. They show that of these 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 gives us an empirical estimate of the average number of rolls needed to get a six. Then we use the power of recursive thinking to find the theoretical answer which turns out to be 6.

This activity is a stretch for grade 9, but it’s a great extension of their work on probability from Grade 8. It also introduces them to the important ideas of recursive thinking.