Puzzle Time - Birthday Holidays

[jon-nyc]

At a certain (fictional) factory, union regulations mandate that any employee's birthday is a holiday for everyone; but except for that, every day is a work day.

How many people should the factory employ to maximize the expected total number of person-days of work in a year?

You may assume that all years have 365 days, all of those days are equally likely to be a given person's birthday, and no two employees are twins (or triplets, etc.).

[Klaus]

Hm, doesn't sound very hard, but I may be wrong.

[Klaus]

click to show

365 or 364 (they yield the same result).

Which yields an expected number of 48943.52 work days.

Why?

The expected number of days that are at least one person's birthday is:

birthdays(n) = 365 - 365*c^n

whereby c = (365-1)/365

That means the expected number of person-days of work in a year is

n  *  (365- birthdays(n))
= 
n * (365-365 + 365*c^n)
=
n*365*c^n

Solving for the root of the derivative of this function yields the desired max at n = 1 / log(365/364) = 364.5.

Maybe there's a simpler way to solve this...

[Klaus]

(making the spoiler invisible)

[Axtremus]

click to show

Away from computer. My intuition says for every person added to the employee pool, it adds one whole person-day to the total amount of working day if there is no birthday-holiday rule, but there is always a positive non-zero probability that the added person's birthday is the same as one of the other employees' birthday, and the expected number of holiday added due to the birthday-holiday rule is always less than one. Therefore the number of working person-day is a monotonically increasing function all the way up to the number of days in a year. Hence to maximize number of working days under the given rule, hire the same number of employees as the number of days in a year.

[taiwan_girl]

I haven't thought about this one yet, but I will.

But I remember when I was in school. And the teacher told us that in a class of ~35 students, the odds that two of the kids had the same birthday were quite high. (I don't remember the exact number, but I want to say that it was somewhere around 30% or so.). Which seems much higher than expected.