# Problem 584: Birthday Problem Revisited

![graphic](img584.gif)

A long long time ago in a galaxy far far away, the Wimwians, inhabitants
of planet WimWi, discovered an unmanned drone that had landed on their
planet. On examining the drone, they uncovered a device that sought the
answer for the so called "Birthday Problem". The description of the
problem was as follows: If people on your planet were to enter a very
large room one by one, what will be the expected number of people in the
room when you first find 3 people with Birthdays within 1 day from each
other. The description further instructed them to enter the answer into
the device and send the drone into space again. Startled by this turn of
events, the Wimwians consulted their best mathematicians. Each year on
Wimwi has 10 days and the mathematicians assumed equally likely
birthdays and ignored leap years (leap years in Wimwi have 11 days), and
found 5.78688636 to be the required answer. As such, the Wimwians
entered this answer and sent the drone back into space. After traveling
light years away, the drone then landed on planet Joka. The same events
ensued except this time, the numbers in the device had changed due to
some unknown technical issues. The description read: If people on your
planet were to enter a very large room one by one, what will be the
expected number of people in the room when you first find 3 people with
Birthdays within 7 days from each other. With a 100-day year on the
planet, the Jokars (inhabitants of Joka) found the answer to be
8.48967364 (rounded to 8 decimal places because the device allowed only
8 places after the decimal point) assuming equally likely birthdays.
They too entered the answer into the device and launched the drone into
space again. This time the drone landed on planet Earth. As before the
numbers in the problem description had changed. It read: If people on
your planet were to enter a very large room one by one, what will be the
expected number of people in the room when you first find 4 people with
Birthdays within 7 days from each other. What would be the answer
(rounded to eight places after the decimal point) the people of Earth
have to enter into the device for a year with 365 days? Ignore leap
years. Also assume that all birthdays are equally likely and independent
of each other.