### 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.

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