The problem is from Introduction to probability 2nd edition byDimitri P. Bertseka> It goes like this:
You go to a party with 500 guests. What is the probability that exactly one other guest has the same birthday as you? Calculate this exactly and also approximately by using Poisson PMF. (For simplicity, exclue birthdays on Feburary 29.)
And in the solution, professor uses $\lambda = np = \frac{499}{365} $ for $P(x=k)=e^{-\lambda}\frac{\lambda^{k}}{k!}$. Isn't $\lambda$ supposed to be $\frac{500}{365}$ because there are basically $500$ trials in this case and each probability is $1\over365$ ?