Facilitate a discussion of the probability answers. Select a sequence of posters to use as examples during this discussion to help all students move from their current thinking up to level 4 (or beyond!).
Level 1:
Students can get the right theoretical answer (or close to it) but the list or display is disorganized, haphazard, or doesn’t clearly show their reasoning.
Level 2:
Students can develop a simulation to answer the question, but their procedure is muddled or they use only a small (less than 50) number of cases.
Level 3:
Students create a systematic theoretical result (with a clear display that shows their reasoning) or simulation (with a clear method and enough cases), but not both.
Level 4:
Students create both a simulation and a theoretical result, but do not connect the two and comment on how well they correspond.
Level 5:
Students connect their theoretical and empirical results.
Issues to raise and questions to ask across presentations
- The theoretical answers should all be the same—1/4. But the empirical answers vary. Why?
- Even though they vary, do the empirical results center on the theoretical answer? (yes, mostly)
- How do the different parts of the theoretical representations correspond to each other? (For example, where does PQ appear in each? And see the four branches in each node of the tree diagram? What do they correspond to in the table?)
- There are several ways to show the theoretical result. Which do you like best?
- There are several ways to do the simulation. Which do you like?
- How well do the theoretical and empirical answers correspond? (This varies.)
- What do you learn from the empirical results that you don’t learn from the theoretical analysis?
This is a tricky issue. It’s easy to think that the theoretical answer—¼—is “right” or “better.” Kids often feel that if they don’t get the theoretical result when they do something that involves probability, it means that their experience is somehow wrong. But it’s not! Random things vary. And at this stage, the empirical trials tell us how much.
Combining Results
To combine all the class simulation results, add the total number of “days” and the total number of “matches.” Then divide. Do not average the percentages! That only works right if every group simulated the same number of days.
Key Idea
You can determine probabilities either theoretically or empirically. If the probability you want is for some combination of events, you need to consider each event separately and combine them.