Dowry Problem In The 1960's

INTRODUCTION

In the late 1950’ s and early 1960’ s there appeared a simple, partly recreational, problem known as the secretary problem, or the marriage problem, or the dowry problem, that made its way around the mathematical community. The problem has a certain appeal.It is easy to state and has a striking solution. It was immediately taken up and developed by certain eminent prob- abilists and statisticians, among them Lindley (1961), Dynkin (1963), Chow, Moriguti, Robbins and Samuels’(1964), and Gilbert and Mosteller (1966). Since that time, the problem has been extended and generalized in many different directions so that now one can say that it constitutes a

”field” of study within mathematics-probability-optimization. One can see

Consider a set of m objects, all of different quality, such that it is always possible to tell which of a given pair of objects is better. Suppose the objects are presented one at a time and at random to an observer, who at each stage either selects the object, thereby designating it as ”the best” and examining no more objects, or rejects the object once and for all and examines another one. For example, observer may be a fussy young lady, whose name is Vanessa and she is 25. She lives in Seattle and likes country music, painting, and swimming. Vanessa would like to marry right man. She has some suitors, but she does not know who is the best

Let B be the event that the last of the i inspected objects is the best of those inspected, and let A be the event that the ith object is the best of all m objects, both inspected and uninspected. Then we want the conditional probability P (A | B) of the event A given that B has already ocurred. According to

P (A | B) =

P (AB)

, P (B) to calculate P (A | B) we need both P (B) and P (AB). Obviously A ⊂ B and hence AB = A, so that P (AB) = P (A). By hypothesis, all possible arrangements of the objects in order of pre- sentation are equiprobable(the objects are presented ”at random”). Hence P (B) is the probability that in random permutation of i distinguishable objects(the objects differ in quality) a given ob- ject(the best of all i objects) occupies the ith place. Since there are i! permutations of all i objects and (i − 1)! permutations subject to the condition that a given object occupy the ith place, this probability is just

P (B) = (i − 1)! = 1 . i1 i

Similarly, P (A) is the probability that in a random permutations of m distinguishable objects, a given object(the best of all m objects) occupies the ith place, and hence

P (A) = (m − 1)! =