Probabilities concerning incoming calls at a call center and a question of independence I work at a call center in a department where we receive two grades of calls: A and B. 15% of our calls are grade A, and 85% are grade B. Only two people work in my specific department, me and my co-worker Dale.
The call center directs calls such that one call is sent to me, the next call is sent to Dale, the call after that comes to me, the next to Dale etc. etc. If Dale does not answer his call, it is sent to his voicemail and does not come to me and vice-versa.
The other day we received 30 calls total. Dale answered 15 of them and I answered 15 of them. Dale received 5 grade A calls and 10 grade B calls. I received 15 grade B calls and 0 grade A calls. I am interested in determining what the chances of this happening are. Is it possible for me to think about my calls separately from Dale's calls altogether? If so, I would imagine that I can determine the probability by taking $.85^{15}$ which is approximately 8.7%.
Also, assuming this independence, I want to figure out how likely it is that this scenario will occur at least once in a month. That is, I am interested in determining how likely it would be for me to have at least one day with no A calls in a 30 day period. Let's assume here that I continue to receive 15 calls a day. Is it possible for me to use this formula:
$1-((1-.087)^{30})$
where 1-.087 is the probability of at least 1 A call. Here I am calculating the probability of at least 1 A call happening 30 times consecutively then subtracting that from 1. This gives me approx. 93.5%.
A few questions I have:
Can I really think about my calls as independent from Dale's calls when determining the probability of the event for me?
Is my method for determining how likely it is for this scenario to occur in 30 days sound?
 A: Yes, based on the information provided, you can think of your calls as independent from Dale's calls. Just think- say you just got an A call. Now the phone is ringing and it is Dale's turn. How does the fact that you just got an A call give you any information about whether the call on Dale's line is A or B? It doesn't. He still has a 15% chance that it is A and 85% chance that it is B. So the calls are independent.
Yes, you correctly determined the probability of having one day with no A calls in a 30 day period.
A: There's two interpretations for this. The first is that each call is truly random, in that there's a 15% chance you get an $A$ call and 85% chance you get a B call. In this case the calls you recieve are independent from Dave's in that if you condition on the number of $A$'s Dale recieved, this still doesn't change the distribution of $A$'s you recieved. 
On the other hand, if you imply that on a given day, exactly 15% of 30 calls, i.e. 4 calls after rounding are $A$ calls, and the rest are $B$ calls, then conditioning on how many $A$s Dale got will definitely change your distribution, i.e. if Dale got 4 $A$ calls, then you have to have gotten 0 $A$ calls. The calls themselves are still independent: both of you have the same chance of getting $A$s or Bs, but once you start conditioning, they are no longer independent. To summarize, if you know a-priori that you guys will receive 4 As and 26Bs, and then you further know that Dave got 4As, then your distribution of As is no longer independent of Dale's. 
I think the former interpretation makes the most sense in which case your formula is fine. 
