What is the better answer to this first year probability test question?
You enter a lottery with $N+1$ people total
$N \sim \mathrm{Poisson}(1)$
A random person is selected to be the winner, what is your probability of winning?
Answer A: Your probability of winning is $\frac{1}{N+1}$ which is a RV from $0$ to $1$
Answer B: Your probability of winning is about 63%
$$ P(W) = \sum_n P(W \mid N = n) P(N=n) = \sum_n \frac{1}{1+n} \frac{e^{-1}}{n!} = 1 - e^{-1} \approx .63$$
My question: What is the better answer here, A or B? In the real world I would be a little hesitant to say 63% because that loses the fact that if there are many lottos then the probability of winning each one could be different. 63% seems more like an expectation which I suppose the total law of probability is an expectation.
It does make sense to me if someone said "$P(W)$ is the unconditional probability of winning" and $1/(n+1)$ is the probability of winning given $N = n$ but it also makes sense to me if someone said the probability of winning is a RV.
I know this is boring question but can you help clear my confusion. Thanks for your help and patience.