One of my first jobs I ever had was being a delivery person in an office. While making deliveries, I always used to wonder : What is the probability today that I will know all the people I am making deliveries to?
I have spent some time trying to convert this situation into a mathematics problem:
- Suppose there is an office. On the first day, the office has $N_0$ people. On the $k^{th}$ day, there are $N_k$ people.
- Each day, there is a $p1$ probability that $n$% of the existing population will join the office (e.g. hiring), a $p2$ probability that $m$% of the existing population will permanently leave the office (e.g. retirement, new job), a $p3 = p2 - p1$ probability that the office population will remain the same as the previous day. We can think of this as the office payroll. (note: the percentages prevent an illogical number of people from leaving the office, i.e. more people leaving than are currently employed)
- On any given day - of the people that are currently on the payroll, there is a $p4$ probability that $j$ of them will not be in the office that day (e.g. sick, vacation)
- On any given day, $q$ % of the current office population will receive a delivery : each person in the office only receives a maximum of 1 delivery. If a person had a delivery and was sick, they will miss their delivery and the delivery for that day will not be rescheduled.
- On a given day - when I make a delivery, I shake hands with the person and remember their name. As soon as I make this delivery, I consider the person as someone whom I have met.
My Question: Assume some fixed values of $N_0$, $p_1$, $p_2$, $p_3$, $p_4$, $m$, $n$, $j$, $q$
- By the end of the $k^{th}$ day, what percent of the current office population will I have met at some point?
- By the end of the $k^{th}$ day, what is the probability that I will know at least 50% of everyone in the office on that day?
I have a feeling that a recursive relation will need to be created to model the evolving office dynamics and deliveries - but I am not sure where to begin. Can someone please help me with this?
Thanks!