*Spoiler for New Scientist Enigma 1747: “Mind your Ps and Qs” *(Follow the link to see the puzzle.)

It’s tempting to say the probability of getting the job is 1 in *PQ*, but no: That’s the probability given no knowledge of what either interviewer said. But we know the second interviewer gave his approval. And then it’s tempting to say, given the second’s approval, all we need is the first’s, and the probability of that is 1/*P*; but no, it doesn’t work like that!

Out of *PQ* applicants, the first will reject (*P*–1)*Q *of them and the second will approve (*P*–1)(*Q*–1) of those; also the first will accept *Q* of them and the second will accept 1 of those. So the total number approved by the second is (*P*–1)(*Q*–1)+1 of which only 1 will get the job.

This means 1/[(*P*–1)(*Q*–1)+1] = 1/(*P*+*Q*) or, inverting and expanding, *PQ*–*P*–*Q*+2 = *P*+*Q*. Then *PQ*–2*P*–2*Q* = –2, or (*P*–2)(*Q*–2) = 2. If *P* and *Q*>*P* are positive integers then we have to have *P*=3, *Q*=4.

So out of every 12 applicants, the first interviewer rejects 8 (2/3 of 12) of which the second accepts 6 (3/4 of 8), and the first accepts 4 (1/3 of 12) of which the second accepts 1 (1/4 of 4): Altogether the second accepts 7, of which 1 gets the job, equal to 1/(*P*+*Q*) as required.