In the town of Ubu (population 365), leap years, weekends, and all traditional holidays have been abolished by mayoral edict.

probability/birthday/ubu New

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Problem

In the town of Ubu (population 365), leap years, weekends, and all
traditional holidays have been abolished by mayoral edict.  However, no
work is done on the birthday of any Ubuite.  How many days a year should
an Ubuite expect to work?

-- David Moews

Show solution

Assuming that birthdays are uniformly and independently distributed
throughout the year, the expected number of different birthdays of the
Ubuites is approximately 230.908, so the people could expect to work
134.092 days per year.

You can compute this as follows:  Let P(b,n) = the probability that n
people have b different birthdays.  Then P(1,1) = 1 and P(b,1) = 0 if
b < 1 or b > 1.  You can compute P(b,365) for b = 1, 2, ..., 365 via
the recurrence relationship:

        P(b,n) = P(b-1,n-1) * (366-b)/365 + P(b,n-1) * (b/365)

Then E(n), the expected number of different birthdays among n Ubuites,
is
                n
            E(n) = sum ( b * P(b,365)
               b=1

-- Dave Dodson <dodson@rsn.hp.com>