Problem: A high school locker room owner has a locker room with 1000 spaces. Each space costs Rs. 100 a day. He has already sold monthly permits to 1001 high school members (knowing that it is likely that not all the high school members would want to keep their bags there at the same time).
If someone with a permit arrives to keep their bag and there are no spaces, the owner will refund Rs. 200 for that day (penalty of Rs.100).
Parking on any given day of the month is independent of every other day.
Has he made the smart decision by selling an extra space?
What I am stuck at is the issue that there are no probabilities given in this question. As we are talking about high school members that means everyone will have lectures on different time of the day (6 working days-from monday to saturday). I think this question will not be requiring any numerical values to be solved. As expected profit is the part of this problem so E[random variable (RV] will be useful in solving this problem. Now If I am not wrong I think binomial RV will also play a major role in getting a general solution for this problem (because if we are interested in that 1 extra member then we will be interested in only two outcomes. 1. if that member will come 2. if that member will not come). That's all I know. I have no idea from where to start. I have no idea how to use CDF(Cumulative distribution function) to my advantage or how to even start this problem and get to the desired results. I am trying to find solution in terms of concepts related to RV not simply probability problems. I would appreciate if anyone can share their thoughts on this problem.