A company has the policy that any employee's birthday is a holiday for the entire company. How many people should the company employ if the expected value of the total number of "people-days" (the product of the number of employees and the number of days worked) is to be maximized?
Answer the question assuming that there are 365 days in a year, that each day is equally likely to be a birthday, and that the employees have no days off except for the birthday/holidays (i.e. no weekends off).
The function to be maximized is n*365*(364/365)^n.
It suffices to maximize n*(364/365)^n.
The first derivative is (364/365)^n + n*(364/365)^n*ln(364/365).
Setting this equal to 0 and solving gives
n = -1/ln(364/365), which is approximately 364.5
So n = 364 or 365.
Based on earlier posts, they are both solutions.
Using Calculus
