Did I do my poisson problem correctly?

Question

If the number of people admitted to the intensive care unit of a 
hospital on any single day is a Poisson random variable with a = 5 
persons per day

b) What's the probability that at least 4 people will be admitted on a 
regular day given at least 2 people have been admitted that day?

ANSWER:
P(X ≥ 4) = 1 - P(X ≤ 3) = 1 - poissoncdf(5,3) = .7349
P(X ≥ 2) = 1 - P(X ≤ 1) = 1 - poissoncdf(5,3) = .9963

P(X ≥ 4 | X ≥ 2) = P(X ≥ 4 ∩ X ≥ 2)/P(X ≥ 2) = (.7349)(.9663)/(.9663) = .7349 <-- final answer

Answer 1

The events $X\geq 4$ and $ X\geq 2$ are not independent so you cannot multiply the probabilities. However their intersection is the same as $X\geq 4$. You should be able to do the rest.

Answer 2

We need $P(X\geq4\mid X\geq2) =\dfrac {P(X\geq4 \cap X\geq2)}{ P(X\geq2)} = \dfrac{P(X\geq4)}{P(X\geq2)}$ since $(X\geq4 \cap X\geq2) = (X\geq4)$.
Now you can solve.