### If you have a question, someone else probably has it too…

I often tell my students, “When you have a question, ask, because you’re probably not the only one who’s wondering.” I heard the same thing when I was a student, but I still felt embarrassed to ask. What if I really was the only one? Wouldn’t I be slowing down class to ask? If no one else was raising their hand, I wouldn’t want to either.

It occurred to me recently that there’s a probabilistic argument that you’re not alone when you have a question: you’re a priori more likely to find yourself with a question shared by many people than one shared by only few, simply because questions that occur to fewer people… don’t occur to as many people! But paradoxically, it’s possible for each student to correctly reason that their questions are probably widely-held ones, yet for most of the questions asked to be unique to the asker.

Let’s make a simple model of my classroom: suppose I regularly make questionable statements, which are either iffy (they cause one student to have a question) or absurd (everyone will ask). Here’s one possible pattern of which students have which questions:

```                Student       What I said:
A B C D E F G H
Question 1        X             iffy
Question 2  X X X X X X X X     absurd
Question 3  X                   iffy
Question 4          X           iffy
Question 5  X X X X X X X X     absurd
Question 6    X                 iffy
Question 7  X X X X X X X X     absurd
Question 8            X         iffy
Question 9              X       iffy
Question 10     X               iffy
Question 11 X X X X X X X X     absurd
Question 12               X     iffy```

I’m doing pretty well—most of my questionable statements are merely iffy. So it’s tempting to think that I should warn my students, “When you have a question, it’s probably only you, so you should just ask me later.” But look at it from student B’s perspective:

```          Student B   What I said:
Question 2    X         absurd
Question 5    X         absurd
Question 6    X         iffy
Question 7    X         absurd
Question 11   X         absurd```

Most of Student B’s questions are about the absurd statements—the ones everyone else is wondering about too! So even though most of the class’s questions are on things only one person is wondering about, each individual‘s questions are much more likely to be about things other people are also wondering about, so it’s still right for them to ask.

Okay, so it’s class time, and Student B asks a question. Should I guess it’s probably not of interest to the rest of the class, because most of the time my questionable statements are merely iffy? Or should I come to the same conclusion that Student B did, that their question is probably shared by everyone?

The answer depends on how likely it is for people to ask a question, given that they have one. If a student only raises their hand once for every eight questions they have, the pattern of raised hands looks something like this:

```                Student       What I said:
A B C D E F G H
Question 1                      iffy
Question 2        X             absurd
Question 3                      iffy
Question 4                      iffy
Question 5  X                   absurd
Question 6                      iffy
Question 7                X     absurd
Question 8            X         iffy
Question 9                      iffy
Question 10                     iffy
Question 11   X                 absurd
Question 12                     iffy```

Then most of the questions actually asked are when I say something absurd, so I should be sure to answer.

As the fraction of the time that the students actually ask their questions gets higher, it becomes easier to tell by the number of askers whether what I’ve said is absurd or merely iffy:

```                Student       What I said:
A B C D E F G H
Question 1        X             iffy
Question 2  X     X   X X       absurd
Question 3                      iffy
Question 4          X           iffy
Question 5  X   X       X X     absurd
Question 6                      iffy
Question 7    X X   X   X       absurd
Question 8            X         iffy
Question 9                      iffy
Question 10     X               iffy
Question 11 X X     X     X     absurd
Question 12                     iffy```

This suggests the following rule of thumb for answering questions:

• If, in general, questions are rare enough that only one person is asking at a time, I should treat every question as if it’s held by more than one person—it probably is.
• If, on the other hand, questions are common enough that several people often have the same question, I can use my discretion to defer answering a question asked by just one person.

Often a class will move from the first category to the second over the course of the term, as people get more comfortable asking their questions. Apparently that makes me less likely to answer them!