This is a soft question
What we [on average] mean when we say "customers/patients visit a hospital $1$ to $3$ times a year on average"?
Does this mean, that the probability to visit $1$, $2$ or $3$ times is exactly $\frac{1}{3}$ ? [discrete uniform]
Or does it mean, that each customer has its own distribution $X_k$, with $\mathbb{E}(X_k) \in [1,3]$ being the only requirement?
When you reduce the question to 'on average $3$ times per year' it becomes clearer. Now the average number of visit equals $3$. Still every visitor has his own number of visits and therefore also his own distribution with $E = 3$.
Saying on average $1$ to $3$ simply gives a range for the expectation but doesn't change the distripution.
In conclusion, statement number 2 is correct.