Hello. I want to know if I am on the right track when solving this problem:

"Assume that customers arrive at a bank in accordance with a Poisson process with rate λ = 6 per hour, and suppose that each customer is a man with probability 2/3 and a woman with probability with 1/3. Now suppose that 10 men arrived in the first 2 hours. How many women would you expect to have arrived in the first 2 hours?"

This kind of problem is relatively new to me, so forgive me if my questions seem ridiculous.

Let's say that A is the event of the number of men arriving in two hours and B are the corresponding events for women.

Are these two events independent of each other?

Is it correct to say that the expected value of total arrivals in two hours are: λ * t = 12 arrivals? Or have I misinterpreted the meaning of λ?

If E[X(2)]=12 is correct so E[women arrival in 2 hours]= 12*(1/3) = 4.

I have a feeling that this was way to easy to solve. Hope some of you guys can give me a hand with this.