6 ms·
If a company gives holidays on people's birthdays, how many people will it hire?
- esrauch 15y agoWhen I read the naive comments that didn't consider the birthday problem properly I scoffed, but I actually think they may have accidentally had some insight that the "correct" answers didn't. The value with the highest expected value of man hours worked would be to hire 365 people, but that might not actually be the best strategy since the variance is going to be pretty high. It's possible that you would be better off with a lower expected value that is also more of a "sure thing" venture
- delinka 15y agoAnd not one of them considered salaries. For example, the answers that said 182 or 183 would get more work for their money hiring 182 employees. But that also assumes everyone makes the same salary. Differing salaries also complicate matters. Adding complexity is generally not the purview of the theoretical mathematician, e.g. "OK, imagine every horse is identical and spherical..."
- jules 15y agoThe story is made up to serve the mathematical problem, not the other way around.
- patio11 15y agoIf you don't immediately knock out a 13 line Ruby script to solve this with a Monte Carlo situation, and the calculus gets heady, try this hypothetical: a) A special act of Congress has made the year only one day long. How many employees would you hire? b) Congress extends the year by one day. Think quick: hire, fire, or no action? c) Congress extends the year by one day. Think quick: hire, fire, or no action? d) Congress extends the year by one day. Think quick: hire, fire, or no action? ... zzzz) Congress extends the year by one day. Think quick: hire, fire, or no action?
- jules 15y agoI don't see how that line of thought helps you solve this problem. Can you elaborate?
- patio11 15y agoIt's the start of a very, very informal argument that the optimum number of randomly assigned objects for any N containers to maximize the number of individually packed containers is N. This problem asks you for what happens if N = 365.
- jules 15y agoIt's just that I don't see the pattern: a) A special act of Congress has made the year only one day long. How many employees would you hire? It doesn't matter, because the people you hire will have their birthday on this day. b) Congress extends the year by one day. Think quick: hire, fire, or no action? Hiring one person gives you 1 workday. Hiring two people gives you 2 workdays with probability 1/2 and 0 workdays with probability 1/2, so the expected value is 1 workday. Hiring 3 people gives you 3 workdays with probability 1/4 and 0 workdays with probability 3/4. So that gives you expected 3/4 workdays. c) Congress extends the year by one day. Think quick: hire, fire, or no action? 1 person: 2 workdays 2 people: 1/3*4 + 2/3*2 = 8/3 3 people: 1/9*6 + 6/9*3 = 8/3 4 people: 1/27*... = ... d) Congress extends the year by one day. Think quick: hire, fire, or no action? That is getting complicated. Obviously I'm missing something here. Can you reveal the insight?
- jules 15y agoWith n employees the probability that any given day is a working day is (364/365)^n. So in a year you get 365n(364/365)^n man working days. Maximizing this continuously gives uniquely n=364.5, so either n=364 or n=365 is the best we can do. Computing both it turns out it doesn't matter which.
- euroclydon 15y agoI would think the probability that any given day is a work day is ((364/365)(5/7))^n
- jules 15y agoWell, if you're thinking about weekends it is (364/365)^n*5/7, so it doesn't make a difference for the maximum.
- deleted 15y ago[deleted]
- ohyes 15y agoDon't hire any employees, just use contractors. You get the added benefit of not having to pay for their healthcare/dental/other benefits.