Date: Fri, 16 Apr 1999 11:52:13 -0600 (CST)
Subject: the birthday problem
Who is asking: Teacher
This is a difficult problem. I am not sure that there is a way to explain the birthday problem that is intuitive. The only way that I see it is from the form of the expression
The key here is that in the fractional part both the numerator and denominator increase with increasing n, but the denominator increases at a much faster rate than the numerator.
Your comment that simplifying the problem by using months (I think of this as a 12 day year) where n = 5 gives p(n) approxiamtely 0.5, made me wonder what happens for "years" of different lengths. If a year is y days long then the appropriate probability, now a function of y and n is
The birthday problem is then, for each y, find the smallest n that gives p(y,n) > 0.5. If this number is called f(y) then the traditional birthday problem has answer f(365) = 23 and your "12 day year" problem gives f(12) = 5. In fact f(y) = 5 if y is anywhere from 10 to 16 days. I did some calculations and produced the following table
This is not an answer to your question but experimenting with the numbers does help me understand why n = 23 is the correct answer to the birthday problem even though it is still counter-intuitive.
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