Problem 584
Birthday Problem Revisited
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.
生日问题回顾
很久很久以前，在一个很远很远的星系，温威（WimWi）星球的原住民温威人发现了一台降落在他们星球上的无人机。他们在检查无人机时发现了一台正在寻找所谓“生日问题”的答案的装置。这个问题的描述是这样的：
如果你的星球上的居民一个接一个地走进一间非常大的房间，第一次出现有3个人的生日两两相差不到1天时，房间中人数的期望是多少。
他们被要求将答案输入这台装置，并将无人机重新送入宇宙。温威人吃了一惊，他们决定咨询他们最好的数学家。温威星球上每年只有10天，同时数学家假设生日在每一天的概率相同，同时忽略闰年（温威星球上的闰年有11天），最终得出了所要求的结果为5.78688636。随后，温威人将答案输入装置，并将无人机重新送回宇宙。
在又旅行了数光年之后，无人机落在了乔卡（Joka）星球上。同样的事件再次发生了，不过这次，装置上的数字由于一些未知的机械故障而发生了变化。这一次的描述写道：
如果你的星球上的居民一个接一个地走进一间非常大的房间，第一次出现有3个人的生日两两相差不到7天时，房间中人数的期望是多少。
乔卡星球的一年有100天，同样假设生日在每一天的概率相同，由此乔卡人（乔卡星球的原住民）得出了答案是8.48967364（保留8位小数，因为装置只给小数点后流出了8个空位）。他们同样地将答案输入装置，并将无人机重新送回宇宙。
这次，无人机落在了地球上。像之前那样，问题中的数字发生了变化。这一回是这样的：
如果你的星球上的居民一个接一个地走进一间非常大的房间，第一次出现有4个人的生日两两相差不到7天时，房间中人数的期望是多少。
地球的一年有365天，忽略闰年，并且假设生日落在每一天的概率是相同且独立的，地球人向这个装置输入的答案应该是多少呢（保留小数点后8位小数）？
（译注： WimWi和Joka分别是印度管理学院艾哈迈达巴德分校和加尔各答分校学生对学校的称呼： WIMWI = Well-known Institute of Management in Western India; Joka则是加尔各答分校所在的城市。）