Suppose you stood on line as part of a group of 30 people waiting to receive a vaccine. After everyone finished, you were told that ten members of the group turned out to have cases of Covid. You never came down with the infection, so you couldn’t have been one of the ten. If you were neither first nor last in line, what is the probability that at least one of the two people next to you in line had the virus (but you got lucky and didn’t catch it anyway) ?

### D19. Covid Conundrum

*Contributed by Arsalan Wares, Valdosta State University*

*This problem originally appeared in the* Prisoner’s Dilemma *in the 2024 Spring issue of the PMP Newsletter. Solutions are no longer being accepted for this Dilemma.*

**Solution**.

Yael Eisenberg (Cornell University), the Eagle Problem Solvers of Georgia Southern, Chris Bistryski (PMP), and Randy Schwartz (Schoolcraft College) wrote in about this one. The condition that you were neither first nor last means that you had exactly two neighbors in line. Since presumably the order of the line was random, choosing your two neighbors is simply a way of choosing two of the other 29 people at random. There are 29 choose 2, or $29\cdot 28/2=406$, ways of doing this. Nineteen of those people were COVID-free, so there are 19 choose 2, or 171 ways to choose COVID-free people. That means in the remaining $406-171=235$ cases, at least one of the neighbors has COVID. Therefore, the chance of this brush with the virus happening is $235/406\approx 57.9$%.