The argument by majority fallacy means arguing that something is true because many people believe it. In the example of the headaches, the argument was that it was likely true because it is true for most people.

What you would want your doctor to do is take the action that maximizes your expected utility, E[U(action)]. Let's simplify a bit, and say that action can be either "do nothing" or "find cause". Then the utilities could be something like:

P(not sick) = 0.99 (most people have nothing to worry about)
U(not sick, do nothing) = 0
U(not sick, find cure) = -1 (unnecessary tests, drugs, worry)
U(sick, do nothing) = -10 (possibly more headaches, or something worse)
U(sick, find cure) = -1 (still need tests, drugs, etc.)

Then:

E[U(do nothing)] = P(not sick) * U(not sick, do nothing) + P(sick) * U(sick, do nothing) = -0.1
E[U(find cure)] = P(not sick) * U(not sick, find cure) + P(sick) * U(sick, find cure) = -1

So with these numbers I just made up, it is better for the doctor to tell you that there is likely nothing to worry about. And you can be pretty sure that in real life, people have done this calculation. Of course in real life there are many more possible actions, such as waiting for a week to see if the headaches go away, which they will likely do if there was nothing wrong. And that is what a doctor will actually tell you to do.