You had to introduce a lot of assumptions to invalidate the n=31. So let me lay it out in more detail.
I recruited 31 white men who self-report having no hair on at least 500cm^2 of their head. They all had hair as adolescents, and lost it gradually throughout their 20s or early 30s. They're all at least 37 now.
I then sort them into the two groups by assigning sequential IDs in random order. Those with an odd ID are in Group A, those with an even ID are in Group B.
I then give each participant in both groups a placebo for 24 weeks, having them check in every 2 weeks to measure hair growth, if any. At 24 weeks, I randomly assign two new courses of treatment to each group. I do this blindly, giving Group A pill X and Group B pill Y. Another 24 weeks (with check-ins every 2 weeks) and I find that Group A (taking pill X) all have full heads of hair. Well, except for Paul and Maury.
Meanwhile, Tom, in Group B (who took pill Y), is the only member of his group to have a full head of hair.
Statistically-speaking, my n=31 was a great sample size, and I'm also pretty sure Tom discovered he had sugar pills and stole the real stuff from Paul or Maury. :)
To address the possibility of seeing these results due to a confounding variable (like above average testosterone), this is about as likely as shuffling a deck of 31 cards, and all the red ones ending up in the front and all the black ones in the back. About 0.0000003%