Finding expectation of a Poisson variable when condition has been made The mall has 3 entrances A, B, C and is open 10 hours a day. The number of people entering the mall at any hour through the entrances A, B, C is Poisson (20), Poisson (30), Poisson (50) respectively. The number of people entering each entrance is independent of the other entrances and independent of different hours. The Question: It is known that 50 people entered the mall in the first hour (from all entrances together). What is the expectation of the number of people entering from Entrance B in the first hour? I tried to break down to indicators (50 indicators each indicator represents whether or not a person entered through B) and it did not work. I also tried the complete expectation formula but it seemed too complicated. The final answer is 15.

Conditional on the total number of events, the events are distributed over the contributing Poisson processes in proportion to their rates; see e.g. Prove that $X|X + Y$ is a Binomial random variable and Poisson random variables and Binomial Theorem. The proportion of $B$ is $\frac{30}{20+30+50}=\frac3{10}$, so the expected number of events from that process is $\frac3{10}\cdot50=15$.