It seems to me that you did it correctly.
What you can plot is probability density versus number of patients. Obviously the number of patients is a discrete variable so you have p(n)=f(n), where n is the number of patients and p(n) is the probaility of getting n patients on any given day.
If you don't yet have a probability density, and only have data points. Then you can plot the number of days that you see a number n, versus n. In other words, let N(n) represent the number of days N that you saw n patients, and then plot N(n) versus n. Eventually, if you get enough data, you can estimate the probalility density as p(n)=N(n)/sum(N), which just says that the probability of observing n patients on a given day is equal to the number of times you previously saw n patients divided by the number of days you saw patients.
In your case you have very limited data. Still, for seriously ill patients you can plot N(n)=1, if n=33, 50, 22, 27, 48, n=0 otherwise; and, for routine patients you can plot N(n)=1, if n=34, 31, 37, 36, 27, n=0 otherwise.