Rigel Kent

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 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 ) 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 sum...

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 X j ( j  = 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 ca...