Lab 1

We will do this in class

Testing loaded dice

Form a group of 3. If students already know people in the class, please ensure each group includes at least 1 person who is new (all 3 should not be from the same study group). Each group will hand their 3-6 modified dice to another group and get 3-6 dice to test. I will provide additional dice that might be loaded (from previous years) or might be fair. Some dice might appear obviously unusual so could be judged to look unfair and loaded - but as you will learn, sometimes appearances deceive!

Groups should prepare a 4-min presentation by one of the group members about their experiment process and results. You get 45 min to prepare. HW#2 will ask for a full writeup of this lab so each group should plan - whether you set one person to take notes or whether you split so different people record different segments.

You should create an experiment protocol before doing any experiment with the possibly loaded dice. Plan how to test them and state what could be learned by particular outcomes. Consider questions like: if dice fair, what is pattern of rolling a 6? Consider and analyze these Possible Protocols before doing experiments.

Possible Protocol #1 (PP1): roll once; if get 6 then conclude it is loaded, if roll any other number then conclude it is not loaded. Analysis of Possible Protocol #1: if it were truly fair, what is chance that it could be judged as loaded? If it were loaded, what is chance that it could be judged as fair?

PP2: roll 30 times; your group can specify decision rules. Consider the stats question: if fair dice are rolled 30 times, what is likely number of 6 resulting? How unusual is it, to get 1 more or less than that? How unusual is it, to get 2 more or less? 3? At least one member of the group should be able to do this with theory; at least one member of the group should be able to write a little program in R to simulate this. Analyze PP2 including the question: if the dice were fair, what is the chance it could be judged as unfair?

PP3: roll 100 times and specify decision rules. Some cases are easy: if every roll comes to 6 then might quickly conclude. But what about the edge cases? Is it fair to say that every conclusion has some level of confidence attached? Where do you set boundaries for decisions? Analyze PP3.

Now devise your own experiment protocol (not ‘possible’ anymore but actual so call it EP1). Analyze it before doing experiments. What is a reasonable number of times to roll your experiment dice? (given time available in class, etc) If you roll the experiment dice a certain number of times and see a particular outcome, then you would conclude it is loaded or not? How confident would you be, in any of those conclusions? Note that while previous PP# emphasized counting just the number of times a 6 came up, you might consider other outcomes such as the average value of the dice when rolled or the distribution of other outcomes (what number is on opposite face from 6? Do you suppose that number might be disproportionately represented if dice were loaded?).

OK now roll the dice and perform the experiment. What conclusion can you draw from your experiment? How confident are you, in this conclusion? How would you revise your protocol if you were to devise EP2? (EP3, EP4 … as appropriate)

Your group’s short presentation should explain justification for your EP1, your experiment results, conclusion and confidence in these results, and what modifications you would make for EP2.

When you are listening to other presentations, you should be alert for good ideas (that you can incorporate into your writeup, with appropriate citation). I trust that you will be supportive of your colleagues.