1. ## Poisson Distribution

Given 400 people, estimate the probability that 3 or more will have a birthday on July 4. Assume there are 365 days in a year and each is equally likely to be the birthday of a randomly chosen person.

I have already tried to solve this problem and got e^-3 but that is not correct. I am unsure of where I messed up in my calculations so if you could show step by step then that would be helpful.

And also, how does that calculation differ from this question:
Calculate the probability that at most 1 person in 500 will have a birthday on Christmas. Assume there are 365 days in a year.

I guess the rate will be $\lambda = \dfrac{400}{365}$
\begin{align*} &\phantom{=}P[\text{at least 3 have birthday on July 4}] \\ &= 1 - P[\text{less than 3 have birthday on July 4}] \\ &= 1 - e^{-\lambda}\left(1+\lambda+\dfrac{\lambda^2}{2}\right ) \approx 0.09876 \end{align*}
for (2) the rate becomes $\lambda = \dfrac{500}{365}$
$P[\text{at most 1 person has birthday on Christmas}] = e^{-\lambda}\left(1+\lambda\right) \approx 0.6023$