Proportions Follow-Up

Earlier this week I was excited to see the problem posted on twitter by Megan Schmidt (@veganmathbeagle) and I wrote about revising my Wednesday plans due to snow and this intriguing problem. I was not the only one thrilled by the problem. Later on Wednesday I saw another post by Joseph Nebus (@nebusj) featuring the same problem! You should definitely check out his post that I linked to and his other writing as well. So, my plan was to play with this problem with my two Discrete Math classes. I borrowed a bag of dice from our new AP Stats teacher and had enough so each student would have four different colored dice. I had each student run twelve trials under two separate sets of rules. One group could manipulate their four dice in any way to try to set up a ratio equivalence. The other group had to decide up front where each color would go. A nice conversation about the problem before we ran trials uncovered some uncertainty about our goals and some debate about whether the additional condition on one group made a difference. It did not take long for us to agree that the group with more freedom should see more success. What happened in each class is what usually happened when I taught AP Stats for five years. Small sets of data do not conform terribly well to theoretical long-term probabilities. I like this in theory but it is frustrating to see the data defy us so often, especially when it feels that the students are really fragile in their understanding of the problem at hand. I had shared this problem with the math team at my school and our AP Computer Science teacher wrote a quick applet and ran thousands of trials. He reported to me that the data he was gathering showed about 6 – 7% success rate with the restriction and about 12% success without. Our data did not match this at all.

Later in the day I saw one of my former students in the hallway. He is a math problem fiend and I figured I would pique his interest by asking him to ponder the problem. He dug in and spent about forty minutes after school with me attacking the problem. Interestingly, he did not grasp my language right away when I spoke about order mattering in one case. When I showed it to him with the colored dice he lit up and immediately attacked my board with renewed vigor. He set up a simple grid and color coded possibilities and we ran through the outcomes that satisfied our requirements. The results we came up with – unfortunately, I did not jot down the numbers – were in line with the results my Comp Sci colleague arrived at in his data. So, I feel like our work was pretty good but it was not elegant at all. Another one of my wizard students suggested using an approach based more in the language of permutations and combinations. I hope to follow up by picking his brain a bit about this. It is almost certainly more elegant than the scribbling we did on the board Weds afternoon.

It was fun to have my brain tickled by a problem that is so easy to approach but so stubborn in giving up an answer. It was also fun to see others on the web tickled by the same problem and now I have another blogger and tweep to keep track of!

Magic in Stats Today

I’ve been feeling grumpy about my AP students lately and I was determined to try and have a serious talk with them about daily diligence and about taking advantage of resources. I also wanted to make sure that I had something positive to do today. So, I tried an experiment that I had read about in the past.

I had read in a couple of places about a stat professor who does an eye opening experiment with her students. I tried it out today and it worked pretty well. I have 20 students in one section and 17 in the other. In each class I formatted a sheet with 93 numbered blank spaces (just how many fit in three columns) and I had 10 pennies in my hand (9 for the smaller class.) I gave them the following instructions:

1) I am going to leave the room for a few minutes. Come and get me when you are done with the following task

2) Randomly distribute the pennies to half the class

3) Those with pennies – toss the penny and record the results until you fill the sheet

4) Those without pennies – imagine you have one and write down the results of your imaginary tosses


When I returned I browsed through the sheets turned in. I was looking for sheets with a few long runs of the same result. I know that in 93 tosses, you are bound to have a couple of runs of 5 or 6 repeats. With the kids that did not have a real penny, I anticipated only runs of three or four at most. In my first class I encountered a student who used his calculator and the result fooled me into thinking he had a penny. Another student who was penniless had a run of seven and fooled me. I did, however, correctly identify three of them. In my second class I was more explicit in my directions about pennies or none (no technology!) and I was five for five in predicting. The students were impressed and the activity resulted in a lively discussion about randomness and unlikely events.

Later tonight, at the dining hall there was an impromptu loud and rowdy round of Happy Birthday for one of our boarding students. 

All in all, a pretty good day.