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McKinney's Generalization of the Birthday Problem: Part II

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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:


McKinney's Generalization of the Birthday Problem: Part I

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In 1966, E. H. McKinney of Ball State University published a paper entitled "Generalized Birthday Problem" in the American Mathematical Monthly, in which he seek to answer two questions related to the famous birthday problem.

The problems considered by McKinney are (a) Suppose n people are selected at random. What is the probability that at least r people will have the same birthday? (b) What is the smallest value of n such that the probability is greater or equal to 1/2 that a least r people to have the same birthday? What follows is the solution by McKinney with some notational adjustment.

Let Xj (= 1, 2, ... , n) be the n random birthdays, let event E' be defined as "r or more birthdays are equal" and the event E be defined as "none of the r random birthdays are equal", then we obviously have P(E)+P(E') = 1, in which P(E) can be computed by summing the probabilities of all ways in which the n random birthdays can take on less than r equal val…