\(~\)

In this note, I will prove that math can be used to derive a formula for love and detail the probabilities of perfect marriages. This is an assignment for English class in which we were instructed to do a project relating to \(\textit{Pride and Prejudice}\), by Jane Austen. Naturally, I incorporated math into this... Hope you enjoy!

\[\text{Mathematical and Statistical Probability of Perfect Marriages in Pride and Prejudice}\]

\[\text{By Trevor Arashiro}\]

\(~\)

\(\textbf{Abstract:}\) Beginning with a hypothetical situation, I will explain the probability that both A) a couple in pride and prejudice will be happiest in marriage B) the possibility of the optimal situation is maximized. This problem is based heavily off of the much more famous problem known as “The Secretary Problem” Of course, since this is a real life problem, mathematics is limited in its power to accurately analyze as is the greatest flaw of \(P_{observed}\) vs \(P_{calculated}\)

## Assumptions

We assume a prior, random distribution with all elements being distinct

The order of the set is both finite and known to all applicants and suitors (this will be explained below)

We will be maximizing \(P_{calculated}\) and not \(P_{observed}\)

Each suitor will chose a wife regardless of her wishes as is customary of the time for parents to marry their daughters off. However, each woman may present herself differently to each man to influence his decision.

Most importantly, the each man's perfect wife defined as \(S_i\) where \(0<i~\epsilon ~\Bbb{N} \leq n\) is 1 distinct element and the set of intersection of all \(S_i\) is \(I(S_i)=S_j\neq S_k\) for \(j\neq k\). The same doesn’t hold for each woman as there are more women than men, thus naturally there is an overlap for women. This holds true throughout the story except for one case which should minimally affect our calculations.

## Terms

\(W(w_i)\) is the set of all women with elements \(w_i\). \(M(m_i)\) is the set of all men with elements \(m_i\)

\(S_i\) is each man's woman of choice and \(T_i\) is each woman's man of choice.

\(I(S_i)\) is the set of every man's perfect wife which is the equivalent to the set \(S(s_i)\). For convenience, I will refer to the latter from now on. The same can be said to the set of every woman's perfect husband.

Characters: 9 Women 6 Men (only 4 get married). Remember that each person is distinct.

## Introduction: Hypothetical Situation

\[\begin{array}{l|c|r}\text{n} & \text{r}& \text{P(r)} \\ \hline 1 & 1 &1.00\\ \hline 2&1&0.500 \\ \hline 3&2&0.500\\ \hline 4&2&0.458\\ \hline 5&3&0.433 \end{array}\] Assume a hypothetical situation as follows:

A man is looking for a single wife to spend the rest of his life with and propagate future generations with. A man and a woman may only be married to one person at a time as these should be monogamous relationships. A house exists where all the women from the story are looking to be married. One male suitor enters and interviews each woman (aka, each set element) one at a time. He will walk from room to room in a random order. He will try to pick the best wife for him knowing only the relative ranking of that woman to the women he interviewed previously and not knowing the overall ranking WRT all the candidates. However, once he says no to a possible wife, he may not recall his decision and is not allowed to choose anyone from before.

## Combinatorics and Total Possibilities

We have 15 possible suitors for marriage, 9 women and 6 men, 4 are to get married. Thus we have a total of \(\dbinom{9}{4}\dbinom{6}{4}=1,890\) possible outcomes and 54 possible couples.

Before we continue, I will try to put this in simple terms through a "function analogy". For each man, there is a best wife and for each wife, there is a best husband. However, due to there being more women than men, we must start thinking in terms of a non-linear function. Define the 9th degree function \(y=f(x)\) where \(x\epsilon W(w_i)\) with 4 real roots, represent the function pairing women with men. Each root of this polynomial represent a successful marriage (the wife loves the husband and the husband loves the wife). Since this function isn't surjective, for each woman, there is only only 1 man right for her. However, for men, all of them have only one right wife for him, but half are best for two women. These are the 5 imaginary roots, women who either aren't right for any man or are right for a man but he does not marry as only 4 of the 6 men choose a wife.

Let's assume that the men chose at random. Then the probability that 1 gets the correct wife is

The probability that each man gets the correct wife is statistically

\[\dfrac{1}{\dbinom{9}{4}}=\dfrac{1}{126}\approx 0.00794\].

Our goal is to maximize this probability and in the worst case have it be \(> \dfrac{1}{126}\).

## Getting Down and Dirty With Math

Some math yields us

#### image credit Wikipedia

Our formula that we obtained is (\(P(r)\) is the probability of 4 successful marriage.)

\[P(r)=\dfrac{r-1}{n}\displaystyle \sum_{i=r}^{n} \dfrac{1}{i-1}\]

Let me try to clear up a few complicated things. Wife \(r\) is the first wife after the stopping point. AKA, wife \(r-1\) is immediately rejected while wife \(r\) is wedded assuming that she is better than the first \(r-1\) applicants.

We need to solve for a ratio between n and r. If we plug and chug values for \(n\) and \(r\), we get.

\[\begin{array}{l|c|r}\text{n} & \text{r}& \text{P(r)} \\ \hline 1 & 1 &1.00\\ \hline 2&1&0.500 \\ \hline 3&2&0.500\\ \hline 4&2&0.458\\ \hline 5&3&0.433 \end{array}\]

Now, say that n tended towards infinity. Let \(v\) represent the ratio of \(\dfrac{r}{n}\) and \(q\) represent the limit of \(\dfrac{i}{n}\). Note that initially, \(q=1\) but as it tends infinity, it approaches \(v\), this is why we have our integral starting at \(v\) and ending at 1. That will maximize the probability. Using some nifty approximations, we can see that

\[P(v)=v\displaystyle \int_v^1 \dfrac{1}{t} \Bbb{d}t=-v\log(v)\]

To maximize this function over the set of reals, we take the derivative and set it equal to 0.

\[P'(v)=-v\log(v) \Bbb{d}v\]

Using product rule for differentiation.

\[P'(v)=-log(v)-1\]

Setting this equal to 0 and solving,

\[-log(v)-1=0 \longrightarrow v=\dfrac{1}{e}\]

We find that as \(\displaystyle \lim_{n\rightarrow \infty}\), the optimal value of \(\dfrac{n}{r}=\dfrac{1}{e}\). In fact, it can be observed that for all \(n\), \(r=\lceil\dfrac{n}{e}\rceil\).

Here, we have \(n=9, 8,7,6\) after each suitor chooses the correct wife for him. Thus we have our probability that each of the four suitors gets the correct wife for him to be \[0.428\cdot 0.414 \cdot 0.410 \cdot 0.406\approx \dfrac{1}{33.9}\approx 0.0295\]

As we can clearly see, we have multiplied our original probability of chosing the correct wife for all by 371%. Thus we have accomplished our goal.

## Comments

Sort by:

TopNewestReally impressive, Trevor!. Some random (cheeky) comments, though, with neither pride nor prejudice ....

(i) I hope that you have an open-minded English teacher;

(ii) Jane Austen is currently rolling in her grave;

(iii) this should be entered as exhibit #1 in "Signs you are a Math Nerd";

(iv) is this the process you will use to find the "correct" girlfriend?

(v) can this be made into a staircase or spiral question?

(vi) Love knows no limits. :)

Log in to reply

Yes, my English teacher is very nice and open minded. When I presented my project topic to her, she said it was original and she was curious as to what I could do.

Unfortunately, she is, but she was one of my favorite writers.

Yes, it should be. Then I can have the #1 and 7 spots.

Unfortunately not. I would need a line of girls chasing after me To use this process :3. My \(r\) value will probably be 1.

Possibly, but I have an idea. Where we make a spiral with summations as side lengths.

Yes, it definately doesn't. Lol, currently doting over a girl whom I can't so much as talk to. Guess that's life XD – Trevor Arashiro · 2 years, 2 months ago

Log in to reply

And being tongue-tied around a girl you're interested in ..... yeah, that's so life. :D – Brian Charlesworth · 2 years, 2 months ago

Log in to reply

– Azhaghu Roopesh M · 2 years, 2 months ago

You always have the excellent open-minded views , always cheerful :DLog in to reply

– Brian Charlesworth · 2 years, 2 months ago

Haha. Yes, I am incorrigibly cheerful. :P Good to hear from you, Azhaghu. I saw a question posted recently that was dedicated in memory of you, and I wondered if something terrible had happened to you. :(Log in to reply

Actually Ishan wanted to post a question for me . I had stopped using Brilliant , but I make some appearances on some days if some friend of mine asks me to take a look at some questions or some notes .

The reason I visited here is the same too . – Azhaghu Roopesh M · 2 years, 2 months ago

Log in to reply

– Brian Charlesworth · 2 years, 2 months ago

Ah, o.k.. Well, I'm sorry that you are injured, but glad that you are alive. :) I realized that you had temporarily "retired" from Brilliant, but when I saw the title of that question I thought something more ominous had happened.Log in to reply

@Azhaghu Roopesh M Sorry for asking you this, but if (and only if) you have the time, could you please respond to a mail I had sent regarding the problem? Thanks very, very, very much! – Ishan Dasgupta Samarendra · 2 years, 2 months ago

Log in to reply

@Azhaghu Roopesh M Looking at what @Jake Lai said, should I remove the integral? That would make the problem shorter and 'neater' to look at (even though the integral is simple enough). – Ishan Dasgupta Samarendra · 2 years, 2 months ago

Log in to reply

@Brian Charlesworth Sir, could you please suggest some changes to me for future problems that I would post? I was quite disappointed as people did not seem to be interested in solving the problem:( Consequently it has a very low number of views:( \[\] I've added the link in case you would please glance at the problem and recommend some changes based on it. – Ishan Dasgupta Samarendra · 2 years, 2 months ago

Log in to reply

– Brian Charlesworth · 2 years, 2 months ago

Well, I prefer to solve problems that are (relatively) simply stated but that end up being more difficult than they appear at first glance. I'm also more likely to work on a problem in which I'm curious about what the actual answer is. So while I can tell that you've put a lot of effort into creating your problem, it isn't one that immediately grabs my attention and curiosity. I tend not to work on questions where the integral is the primary focus of the question; I may have when I was younger and just learning about them, but now if an integral comes up that I'm not immediately familiar with I tend to just look it up in a table or use WolframAlpha, which is effective but not much fun. I'm getting lazier as I get older. :)Log in to reply

– Ishan Dasgupta Samarendra · 2 years, 2 months ago

Alright Sir, thanks a ton for your inputs! But Sir, just coming back to the same point, that integral was by no means the main focus of the problem. The main focus was on maximizing \((p^{333}+q^{333}+r^{333})\) subject to a condition which could be derived from the integral. Sir, according to you, should I remove the integral from the problem and simply supply the required condition?\[\] Ps. If this is your lazy version, how I wish I could just get a billionth of your younger version! That would be more than enough to see me through all the academic struggles that would come later on in my life!Log in to reply

P.S.. You're solving much more complicated problems than I could at your age, (I didn't even know about calculus at 16), so it looks like you are well-prepared for your future. :) – Brian Charlesworth · 2 years, 2 months ago

Log in to reply

It does look much cleaner now @Jake Lai Very happy that the Geometry problem interests you, Sir. Thanks a lot Sir! – Ishan Dasgupta Samarendra · 2 years, 2 months ago

Done Sir.Log in to reply

I wrongly entered an answer to increase the points to 370 and then answered it correctly . – Azhaghu Roopesh M · 2 years, 2 months ago

Log in to reply

– Ishan Dasgupta Samarendra · 2 years, 2 months ago

You know, I was about to ask you how you had solved it, and then was about to add the method I thought would work! Due to some sense I managed to stop!Log in to reply

– Ishan Dasgupta Samarendra · 2 years, 2 months ago

Started laughing when I read this!Log in to reply

I only learnt Calculus last year . I used to spend loads and loads of hours on the Internet trying to get better and better .

Btw , are you on any social networking site ? I use Google Plus to discuss academic doubts , it'd be great if you had an account in it . – Azhaghu Roopesh M · 2 years, 2 months ago

Log in to reply

– Brian Charlesworth · 2 years, 2 months ago

No, my online "networking" is restricted to Brilliant. I prefer not to have too many distractions. :)Log in to reply

– Jake Lai · 2 years, 2 months ago

I'm sorry if it seems rude or out of place, but your problems are rather tedious; many people are loathe of doing such questions, myself included. Make a simple but difficult problem!Log in to reply

@Jake Lai @Brian Charlesworth @Trevor Arashiro Please could you tell me if this is better? – Ishan Dasgupta Samarendra · 2 years, 2 months ago

Log in to reply

– Ishan Dasgupta Samarendra · 2 years, 2 months ago

Yeah, I was thinking the same thing. But actually the integral in this question is quite easy (it just looks tedious at first). The maximising part becomes interesting (to me).Log in to reply

@Brian Charlesworth – Ishan Dasgupta Samarendra · 2 years, 2 months ago

No, Sir, that was not the reason I had posted the problem with that title. I had promised I would post a question in his name since he had left Brilliant and I felt really bad as he was (is) my best friend here. Very nice list of comments Sir! I really liked them!Log in to reply

I love how Baye's Theorem fits into this! And its limit resembles a derangement case. Love your note! Keep it up Trevor, looking forward for more of your notes. – Pi Han Goh · 2 years, 2 months ago

Log in to reply

– Trevor Arashiro · 2 years, 2 months ago

Thanks Pi Han! Really appreciate your and other's support :)Log in to reply

Having mathematics even in marriages is super interesting and I think you are correct in your efforts. Hats off! – Nihar Mahajan · 2 years, 2 months ago

Log in to reply

– Trevor Arashiro · 2 years, 2 months ago

Thank you for your support Nihar :). I'm very relieved that every one liked the connection.Log in to reply

Well, you have wrote an awesome note, mate!

Maths+Loveis always awesome! – Pranjal Jain · 2 years, 2 months agoLog in to reply

Lovehistory ? ;) – Azhaghu Roopesh M · 2 years, 2 months agoLog in to reply

– Pranjal Jain · 2 years, 2 months ago

Haha! No comments as of now. ;)Log in to reply

Great Note!Nice work Trevor – Kalpok Guha · 2 years, 2 months ago

Log in to reply

– Trevor Arashiro · 2 years, 2 months ago

Thanks for your appreciation :)Log in to reply

Great. – Krishna Ar · 2 years, 2 months ago

Log in to reply

– Trevor Arashiro · 2 years, 2 months ago

Thanks.Log in to reply

Hey!

This is a really innovative way of presenting things . I feel as if I should try out an essay this way too . (Although I very well know that I cannot present it to my teachers at school as I know what their response to it will be !)

I hope you won't mind if I share your work with my friends :D

Keep up the good work ! – Azhaghu Roopesh M · 2 years, 2 months ago

Log in to reply

I would be honored if you shared it with your friends. That would mean so much to me. – Trevor Arashiro · 2 years, 2 months ago

Log in to reply

– Azhaghu Roopesh M · 2 years, 2 months ago

Of course I did share it . Do you want me to share it on G+ ? But all my frnds from there have already seen it .Log in to reply

– Siddhartha Srivastava · 2 years, 2 months ago

Isn't your school over? Also how'd the Mains go?Log in to reply

Jee Mains was bad . I was expecting 262 but due to the SILLY errors on the part of those INTELLIGENT guys who made the question paper and the answer key , I'm getting less than 250 :(

This is one of the few reasons I actually hate JEE .

But I know I'm talking with JEE 2016 AIR 1 ,right ? :D – Azhaghu Roopesh M · 2 years, 2 months ago

Log in to reply

And give them essays? Most people don't want to submit their essays while they're in school, and you're talking about submitting essays when you're not? xD

and Ouch. That sucks. The fact that the entrance exam for the most prestigious colleges in India contains mistakes this frequently is alarming enough; now they're not even correcting those mistakes properly. And don't worry, Mains was pretty easy. You'll get awesome marks in Advance.

And introduce me as well to the AIR 1 when you get the chance. :P – Siddhartha Srivastava · 2 years, 2 months ago

Log in to reply

Note: this isn't finished and I rushed some calculations, so they're not all right. – Trevor Arashiro · 2 years, 2 months ago

Log in to reply

Total number of outcomes is \( \binom{9}{4} \binom{6}{4} 4! \). After choosing the 4 women and men to be married, you have to choose who marries whom, which gives you \( 4! \) more ways. This also means that the probability that every man is married to the woman of their choice is lower by a factor of \( \frac{1}{24} \)

The probability you've given of exactly 1 man getting married to the right woman is not correct. It most likely involves some sort of derangement. Likewise for the case of 2 men.

You haven't actually defined a lot of variables. For example, you don't define what \( P(r) \) or \( n \) is. You also don't really mention explicitly what your strategy is.

Log in to reply

(ii) would it be correct if I said at least 1 correct marriage?

(iii) I define \( n\) above. I'll add the definition of \(P(r)\) – Trevor Arashiro · 2 years, 2 months ago

Log in to reply

That would still involve derangement, unless I'm missing out on some simple solution. You would be better off removing that bit.

One last thing. You've mentioned the woman's choices several times, but the secretary's problem doesn't consider the woman's choice. So there is no point of mentioning their choices.

Also, have you seen this? Watch the full thing. – Siddhartha Srivastava · 2 years, 2 months ago

Log in to reply

@Trevor Arashiro . – Kalash Verma · 2 years, 2 months ago

Haha😁😀 Hardwiring math and English. It's inventive and a great paper. Hope your teacher doesn't get annoyed.Log in to reply

– Azhaghu Roopesh M · 2 years, 2 months ago

I know my teacher will be irritated to the helm , she would see it as something that pollutes the essence of Writing Skills . But you wouldn't want me telling you how innovative things are treated in our schools ,do you ?Log in to reply

– Kalash Verma · 2 years, 2 months ago

No , I know quite well. And Of course its innovative and A GREAT WORK .Trevor!!Log in to reply

soul mate decided in heaven already the only duty of us is to fill his/her life with lot of love amd happiness. then the probability will be maximum. – Aayush Shrivastava · 2 years, 2 months ago

Log in to reply