How to calculate the propabilty of a specific Y in a poisson regression model in R?

Question

In my R session, I fitted a poisson regression model on patients in an emergency room like this:

fit <- glm(Patients ~ Day + Event, data = df, family = poisson)

(Day is day of the week; Event is "yes"/"no" whether there is a public event in town, like a concert or football match)

Now I can predict Y (the count of patients) for specific Day and Event like this:

predict(fit, data.frame(Day="Tu", Event="yes"), type = "response")

which gives me 12.8. So we can expect about 13 patients on a Tuesday when there is a public event in town.

My question is: I would like to calculate the propability for Y=16 patients on a "Tuesday", based on my fitted model. How would I do that?