Poisson processes of two airline companies. (two independent Poisson processes) Easyjet and  KLM  planes  request  landing  permission  at  Heathrow airport  according  to independent Poisson processes with intensities $\lambda$ and $\mu$ per hour, respectively for Easyjet and KLM.
The following questions are about this process:

a) Given that exactly three Easyjet planes have requested landing permission over the course of a two-hour period, what is the probability that at least one of these requests occurred during the first hour of that period?

This is a conditional event. I am having serious trouble thinking how to write this down however... which makes me feel rather stupid.

b) What  is  the  probability  that  among  the  first  three  requests  for  landing  permission exactly two are generated by KLM planes?

Two of the events should be KLM events, the other one is an Easyjet event, this probability is given by:
$$ \binom {3}{2} \frac{\lambda}{\lambda+ \mu} \cdot \left( \frac{\mu}{\lambda + \mu} \right)^2.$$
 A: The first one is a fact that is generally taught in class and not expected to be derived (based on my limited experience searching for similar courses). Were you not taught that the arrival times are i.i.d. uniform on $[0,T]$ conditioned on the number of arrivals up to a time $T$? It is 2.4.3 on Durrett EOSP (available online free). The proof of it is not bad; you just write out the conditional, and note that it does not depend on the exact values of the arrival times (Normally, the joint density of the $k$ arrival times only depends on the last one, and here, all we know is that the $k+1$th arrival is larger than $T$). With this knowledge at hand, I am sure you can do that first part.
Your second one is exactly right.
EDIT: Forgot the word i.i.d., so added that, and adding a couple guiding questions for the first part:
a) Say the names of the planes that landed are A, B, C (not necessarily in landing order, so they could have landed A - C - B, say) What is the probability that $B$ landed in the first hour?
b) What is the probability that both $B$ and $C$ landed in the first hour?
