There's no simple "right" distribution for the process because the actual process is not simple. There's dependence between events and variation in rate over time (as jbowman mentioned already in comments), but the question doesn't provide information on those aspects of the process.
For an interview question the "right" distribution to choose is not what is correct for the process but whichever distribution the interviewer thinks is the right one.
I'd guess the intent is actually to assume a Poisson process (since that allows a numerical answer). I wouldn't be able to give that answer without pointing out why it was wrong, however. I'd attempt to explain the directions in which the answer might move as other - possibly more realistic - assumptions were used. There's not enough information to give a good numerical answer though.
Of course they might actually be looking for three tiers of answer (those with no idea how to proceed, those who can recognize they're trying to set up a Poisson process model, and those who can see that and explain why that's wrong).
[I'd harbour some reservations about working at a place where they thought that just using the Poisson would be a good answer. A question like that at least potentially reveals as much about the employer (or the quality of the work done there) as it does about the employee. If I were interviewing for more than one job, other things being equal something like that could be enough for me to lean towards choosing someplace else.]