Shared alien birthdays
On the planet Maath, each year is exactly 20 days long. Find the smallest positive integer n such that if n randomly selected Maathians were in the same room, it would be more likely than not that two of them would share the same birthday. Assume that the birthday of each Maathian is independent and uniformly distributed among the 20 days of the year.
Inspired by this comic .
The answer is 6.
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2 solutions
I was guessing 8 or 9 before 'doing the math'. Yet another unexpected result in Discrete Mathematics. :)
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Number of people required ∝ d , where d is the number of days in the year, so I guessed 5 instead (based on an answer of 23 for 365 days).
Consider instead the complementary probability that none of them share a birthday.
The first person could have any birthday (probability 1) and the second could have any of the 19 remaining available birthdays (probability 2 0 1 9 ). The 3rd person could have any of the 18 remaining available birthdays (probability 2 0 1 8 ). And so on.
So we calculate: 1 × 2 0 1 9 × 2 0 1 8 × 2 0 1 7 × . . . , stopping when the answer first gets below 0.5.
This happens for 1 × 2 0 1 9 × 2 0 1 8 × 2 0 1 7 × 2 0 1 6 × 2 0 1 5 , so the answer is 6.
[This is equivalent to Andy's solution but might be easier for some to follow...]
Consider the complement probability: the probability that all n Maathians in the room have different birthdays. One can establish a bijection between the set of distributions of birthdays of n distinct Maathians and the set of distributions of n objects selected with replacement from 20 objects.
The number of distributions of n objects selected with replacement from 20 objects is 2 0 n .
The number of distributions of n objects selected with replacement from 20 objects such that each object selected is different is ( 2 0 − n ) ! 2 0 ! .
Thus, the probability that at least one pair of Maathians among n share a birthday is:
p = 1 − 2 0 n ( 2 0 − n ) ! 2 0 !
Testing values of n until this probability is greater than 2 1 gives n = 6 . This gives a probability of p ≈ 0 . 5 6 3 9 5 that a pair of Maathians share the same birthday.