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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!

stats_noob
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    I recommend you start with something a lot simpler – e.g., a fixed number $N$ of people in the office, one delivery every day with everyone in the office equally likely to be the one getting the delivery; what's the expected number of people you've met after $k$ days? What's the probability you know at least $50%$ of your coworkers after $k$ days? If you can figure those out, then you can add complications, one at a time. And if you can't figure the simple problem out, you'll have some idea of just how complicated a question you are asking! – Gerry Myerson Jul 14 '23 at 07:35
  • @ Gerry Myerson: Thank you for your reply! I have been working on this problem for a few weeks now and have been approaching it in smaller steps ... trying to build up to the final question (i.e. the current question I posted). Here are some intermediate problems I tried working on: – stats_noob Jul 15 '23 at 17:06
  • https://math.stackexchange.com/questions/4730737/expanding-a-probability-tree – stats_noob Jul 15 '23 at 17:06
  • https://math.stackexchange.com/questions/4730130/probability-of-a-random-variable-falling-between-two-ranges – stats_noob Jul 15 '23 at 17:06
  • https://math.stackexchange.com/questions/4727991/expected-score-in-a-coin-flipping-game – stats_noob Jul 15 '23 at 17:07
  • I am trying to see if I can somehow use the logic from these problems to solve my current problem ... the main difference between the current problem and the problems in these links that I posted : in my current problem, we start "tracking" the individuals in the population as well as the overall size of the population - in the earlier questions I was asking, only the overall size of the population was of interest. Thank you so much for everything! – stats_noob Jul 15 '23 at 17:09
  • this seems similar to the birthday problem – Mark Jul 20 '23 at 03:18
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    Hardly anybody will jump to solve a problem with nine parameters, bounty or no bounty. – leonbloy Jul 21 '23 at 02:00
  • @leonbloy: i guess this is where you use simulation based approaches... – stats_noob Jul 21 '23 at 03:01
  • You start by asking, "What is the probability today that I will know all the people I am making deliveries to?" but then your question is about people in an office. So I'm confused. Also, there's too much stuff going on in the question and you should ask simpler questions first. – Adam Rubinson Jul 26 '23 at 21:25
  • Now posted to MO, https://mathoverflow.net/questions/451536/probability-of-knowing-someone-in-the-office without notifying either site of the post to the other. – Gerry Myerson Jul 28 '23 at 04:32

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