### 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

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

Following McKinney, we define the general form of the probability

It follows that

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:

…

*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

*n*1 +*n*2 = 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*;*n*1,*n*2) asIt 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:

…