In the math class I taught today I was asked a question, and I was unable to give a good answer.
The problem was as follows: A certain factory produces throat tablets. In each pack, there are from 48 to 52 tablets, with a given probability distribution in a table (I don't remember the exact probabilities; it was far from uniform). If we take two of these packs, what is the probability that there are 102 tablets in total?
The provided solution (as well as how I would've solved it myself) went on to say that there are three ways that this can happen:
- The first box has 50 and the second box has 52
- The first box has 51 and the second box has 51
- The first box has 52 and the second box has 50
For each of these three probabilities, the probability is calculated by multiplying the relevant probabilities. Then the three results are added up and that's the answer.
One student asked me why there are three cases and not two. Why are the first and third cases separate? Why is there a first box and a second box, and not just two boxes?
Another, more standard, example is when working with the sum of two thrown dice. Same thing here: Why can we declare that there is a first die and a second die (or a blue die and a red die)?
I know that we can "pretend to change" the problem text to say that we pick first one box and count the tablets, and then pick the second box and count the tablets. But the problem as stated doesn't say that. According to the statement the two boxes are entirely equal. How can I justify breaking this symmetry? (Apart form saying "See, it gives the answer you're supposed to get.")
Note that this is not about the similar question in combinatorics. When asking "how many ways are there to select three students from the class to form a committee", or something like that, I think many of them have understood (and I feel more confident in how to explain) when order matters and when it doesn't in such questions. And I am similarily confident in the probability questions that spring directly out form such scenarios ("What is the probability that Alice and Bob are both chosen as part of the committee?").