Well, couple of months ago- I had posted a topic trying to prove why a person cannot be lonely- and termed it as the Loners Pair Dependency Principle. However, recenlty i was contemplating on what would be the probablity of a person finding that "pair". Just as everything seemed placed to perfection , so were the thoughts- and the Result - "Pair Probablity Theorem"..Lolz!. Sorry for follwing poetries with yet another crappy thoughts..:)- Hope you like this though.

You are seeking a spouse and, obviously, want to find the best match possible. As you meet and date “candidates”, you have the opportunity to determine how well matched you are as a couple. There are several rules to this dating game:

· It is generally considered bad form to date seriously two different people simultaneously, so you consider each person one at a time.

· You can date someone for any length of time, but eventually, you must either “select” them or say “no”, and move on to another candidate.

· Once someone has been passed over, you cannot go back. No is forever.

· If there are N candidates, how can you maximize the probability that you select your best match?

It is essential that you know when a candidate is a good one and when they are not so good. The only way to gain some understanding of what is “good” is to “play the field”. Date several people without serious intent to determine what attributes are important to you. This is similar to the baseball strategy of “taking a strike” before hitting. Taking a strike gives the hitter the opportunity to better judge what is a good pitch from this pitcher. In this model, we will employ the “play the field” or “take a strike” strategy.

Strategy for Finding a Spouse: Date k people without making a selection. Then, select the first person judged to be better than any of the first k.

What is the relationship between N and k that maximizes our probability of selecting the very best spouse from N choices. If k is small, we have little information. Without sufficient information about the quality of the choices, we can make a hasty and unwise uninformed choice. If k is large, then the very best choice has a greater probability of being among the first k, which guarantees that our selection will not be optimal. This, then, is the max-min dynamic. As k increases, we can make a better and better choice. But as k increases, e face the likelihood that our best choice has already passed us by before we begin the selection process.

A Mathematical Model

We want to find the value of k (relative to N) that gives us the greatest probability of selecting the best spouse for among the N potential choices. We will develop a function P(k) that will compute the probability of success as a function of k. Remember, k is an integer, so the domain of this function will be k= 0,1,2,...N-1. If k=0 , this is equivalent to selecting the first person and if k=N-1 , we select the last person.

To define P(k) , we consider the possible location of the best choice. They could be anywhere from 1 to N. We will be successful if we select the best person, otherwise we are unsuccessful.If we let k go by and then select the first person better than any of the first k, the probability of success can be computed using the diagram below:

The best person could be in the first or the second or the third, or, in fact, any position in the list. So the probability is the sum of the individual probabilities of being in a position, and being selected.
So ,

Calculating the Probabilities

What is the probability that the best person is in Position 1? The best person is equally likely to be in Position 1 as any other position. All positions are equally likely, so each has probability p=1/N. Now, if the best person is in Position 1, what is the probability that they will be chosen? Since we will not choose any of the first k, this probability is zero. So the combined probability of being in the first position and being chosen is p=(1/N).0=0..

In fact, for the first k positions, the value of P is 0.

So, we have

The first non-zero term in P comes from Position (k+1) . The probability that the best choice is at this position is again p=1/N . If the best choice is at this position, the probability that it will be selected using this procedure is p = 1. At this point,

While we cannot solve the problem directly using calculus, we can generate an approximation using calculus. Students in calculus are familiar with the principle of using discrete models and methods to approximate continuous models. They see this when using Euler’s method to generate approximate solutions to differential equations and when they use Reimann sums, or the Trapezoid Rule to approximate a definite integral. In this problem, we will do the reverse, We have a discrete function and we will approximate it with a continuous function. By using the more powerful techniques of calculus on the continuous approximation, we can learn something about our discrete model.

The probability of success settles down as k increases to approximately 0.368 as well. Using this process, we find that we can be successful in selecting the best from a group of N by letting approximately 37% of the available positions go by, then selecting the first choice better than any seen before about 37% of the time. And this is true no matter how large N is! This is a strikingly high probability. Using this process, you can select the best out of 5000 almost 37% of the time by letting the first 1839 go by and then selecting the first choice better than any of those 1839.

It also suggests to students that marrying your high school sweetheart is not a particularly good strategy. Don’t get too serious too soon. Go out with a number of people to see who you like and who likes you. Then make your choice.