Sunday, May 23, 2010

McKinney's Generalization of the Birthday Problem: Part II

In Part I of the article, we considered the expression for computing the probability that at least r people will have the same birthday, given n people are selected at random.

In this article, we will consider a special case to illustrate the use of formula numerically: Suppose five people are selected at random, what is the probability that at least three people will share the same birthday?"

In this example, the Frobenius equation n1 + n2 = 5 has three sets of solutions and they can be tabulated as follows:


Following McKinney, we define the general form of the probability P (n; n1, n2) as


It follows that P(E) can be computed as follows:


The required probability that at least three people are sharing their birthdays is therefore:


Alternatively, we could also arrive at the same result if the problem is approached from another direction, but we would require a slight modification on McKinney's formula.


The required probability is thus the summation of the following fractions:


This is approximately 0.007 percent. From the probability table above, we noted that if we were to select five people at random, chances are 97.3% that they will all have unique birthday, this is consistent with our intuition.

2 comments:

Anonymous said...

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Anonymous said...

I read this article, useful to me, thank you for writing it and put it online.
I'm trying to write an article on the same subject.

reabiagio.wordpress.com

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