Two probability problems that seem similar but have different answers

Earlier in the week we looked at the game Ox Blocks which uses a 6-sided die with 2 sides each having O’s, X’s, and a blank. The game is a really fun version of tic tac toe:

Playing with Ox Blocks thanks to the Mathematical Objects podcast

Playing this game reminded me of an old project we’d done on a fun probability problem from Elchanan Mossel:

Exploring Elchanan Mossel’s fantastic probability problem.

For today’s project we looked at two problems inspired by these two projects. The problems seem pretty similar:

(1) If you have a fair 6-sided die with sides marked 2, 2, 4, 4, 6, and 6, how many rolls on average will it take for you to roll a 6.

(2) If you have a fair 6-sided die with sides marked 1, 2, 3, 4, 5, and 6, how many rolls on average will it take to roll a 6 if any sequence of rolls containing an odd number prior to seeing a 6 doesn’t count. So, 2, 4, 4, 6 would count, for example, and 2, 4, 5, 6 would not count.

I started the project today looking at the first problem, which is inspired by the Ox Blocks project:

Now we moved to the 2nd problem. To introduce the problem I had the boys play the game a few times and we found that lots of sequences of rolls were thrown out:

To help the boys understand this second game a bit more I moved to a slightly different question -> for valid sequences of rolls in the 2nd game, how often do you see a 6 on the first roll.

This question was slightly difficult for the kids to understand, but we made pretty good progress:

Finally, we went to the computer to run a simulation for the 2nd game. This video runs a little long as I asked my younger son to explain the program. But once we get through the explanation we see that their guesses for the expected number of rolls and also the percentage of 6’s on the first roll were roughly right!