RATIO OR FRACTION?

This might not be in any book, but it is quite intuitive and has to be true.

At least sometimes we need to fix the books.

Ratio has to do with proportions and fraction has to do with parts of a whole.

If we go for etymology, ratio is associated with reckoning (http://www.etymonline.com/index.php?term=ratio) and fraction is associated with breaking and part (http://www.etymonline.com/index.php?allowed_in_frame=0&search=fraction&searchmode=none), so that everything is compatible with what we are saying. 

There is then a huge difference between one item and another.

We use ratio, and represent it by a:b, when we mean proportion and we use fraction, and represent it by a/b, when we mean parts of a whole.

The media frequently changes normal figures of entrance exams into proportions, what may cause a lot of misconception and misunderstanding even in our experienced scientific minds.

http://blogs.abc.net.au/antonygreen/2013/11/average-candidates-per-vacancy-at-federal-elections-1949-2013.html talks about an average of 13.2 candidates per vacancy when they had 529 candidates contesting 40 Senate seats.

It then looks like they have simply divided 529 by 40, is it not?

If we put 529/40 in our calculators, we certainly get 13.225.

What is wrong with this? – you will ask.

We also initially saw nothing wrong with that sort of assertion.

This assertion is not only untrue; it is also absurd.

We cannot make the three-point rule with this sort of matter because it is not true that there is a proportion involved there. 



See: 529 candidates per 40 vacancies does not imply that if we have fewer vacancies (say 30), we will have fewer candidates or that if we have fewer vacancies, we will have more candidates, or that if we have more vacancies (say 50), we will have fewer candidates or that if we have more vacancies, we will have more candidates.

Even if there were a projection in this direction, say that the number of vacancies could influence the candidate’s decisions, we would need to know how exactly that happens to be able to tell whether we have a proportion or not.

It is not true that, if we double the number of seats, we have twice as many candidates or half as many. 



Yet, with true proportions, that is an acceptable inference.

We will have the same number of candidates if the number of our vacancies changes after everyone has entered the competition, so that the dispute over one or 40 seats involves always the same 529 candidates and we then have 529 people disputing over the 40 vacancies all together, at the same time.

The upshot is that it is wrong making these divisions and telling people that things are like that.

It is not because we have these figures (529 and 40) that we are disputing with at most 13 other people over one vacancy in the Senate. 



No, we are still disputing them with everyone else, that is, with the other 528 candidates.

Notice that we could only do things the other way around: suppose that all vacancies have been assigned apart from one. We then have 490 candidates for one vacancy. 

All that we say and write has got a certain level of social impact (Psycholinguistics) and it is really not pleasant feeling lured, so that we should not make these outrule divisions anymore (we recently saw a fellow using the term illegal in this situation and we felt tempted to do that, but we then thought twice and decided for inventing outrule (copying outlaw), since what we mean is that it is against the rules of Science/Mathematics).

________________________________________________________

DO YOU WANT TO PLAY LOVE GAMES? GET THE BOOK HERE!

_________________________________________________________

    

The Unexpected Hanging Problem

Today we had contact with the Unexpected Hanging Problem (http://lesswrong.com/lw/3zd/resolving_the_unexpected_hanging_paradox/) because of LinkedIn (Math, Math Education, Math Culture, A Logical Reasoning Problem, discussion proposed by Prof. Dr. Fuchun Huang).

Interesting enough that nobody seems to take notice of the most basic things in this type of problem.

We must assume that  we are indeed Very Important People (VIPs), we reckon, and we must then say that at least the Who’s Who Marquis and the IBC had the bravery and the wisdom of printing that we are such in their books, which list our most important personalities. 

In plain terms, this problem is quite simple.

Yes, what they write is true. Oh, well, it is also untrue, like part to one side and part to the other.

Basically, If we are on a Thursday before 4:30 PM, say, supposing that it takes us 30 minutes to get it all done, from picking the victim at the cell to hanging, then the hanging can still happen on the same day and it will surprise us because it might happen on the own Thursday or on Friday, according to the variables we have been given.

The same will happen with all days of the week, trivially, and this is, according to Science, the right way to see it. Moreover, because there is surprise until 4:30 PM of Thursday, then there is surprise on any day of the week at any time of the day before that time... .

Now, assuming that the system only hangs from 8 AM to 5 PM, and that hanging takes 30 minutes, like the entire process, which starts at the moment in which the prisoner is picked, and that the staff never works, like those who deal with the hanging process, in any hypothesis, after 5 PM, then the victim can be hanged at any day of the week until 4:30 PM, apart from Friday, and it will always be considered a surprise (like perhaps apart from holidays and other atypical times).

____________________________________________________________

DO YOU WANT TO PLAY LOVE GAMES? GET THE BOOK HERE!

____________________________________________________________

    

The Birthday Problem: Another moment of shame for the scientific community

This problem has appeared in the press because of a few mathematicians, some of them considered important. 


The last mathematician we watched speaking about it on TV was a professor from an English university, and that was on Australian TV.  


We find http://theconversation.com/the-birthday-problem-what-are-the-odds-of-sharing-b-days-16709, however, mentioning the result in the same way. 

It is all nonsense and it all looks like a conversation of non-mathematicians.

Basically, probability brings us only the chance that something happens.

An event that has a chance x of happening, say a chance of 1 in 1000, may happen or not: We may throw the same ball on a field marked with one thousand numbers, for instance, and never get it to land on number 1, not mattering how many throw the ball or for how long. Yet, if asked, we will say that all the people who threw the ball had 1 in 1000 of chance of getting it to land there, on number 1. 

Suppose that we had 1000 people throwing it today. Suppose we never got a single one to land the ball on number 1.

Does that defy the laws of Mathematics?

Were we all nonclassicists, like Graham Priest, we would probably get it all mixed up in the basics, and we would then start believing that the mathematical foundations are all nonsense... . 

If 2000 people come and throw the ball and none of them gets it to land on number 1 even trying very hard to get that, what can we infer? One would say: There might be something wrong with the field because 2000 have tried and nothing. That is a bit too much!

Was everyone targeting number 1? 

Perhaps it is the weight of the ball or something... .

Were it a computer program, we could say that the program had been built in such a way that nobody would succeed in doing that, for instance: It is possible that the field does not allow for the ball to land there,... . It is also possible that the ball be specially prepared so that nobody get it.

It is also possible that we can predict the strength of the throw through body dimensions, for instance, and we select the people who throw the ball based on that, say nobody that can physically do it gets to be selected.

It is also possible that all is honest and nobody can actually do it, like ever.

One would definitely be forced to believe however that there is something dishonest with the ball-throwing thing by the 1000th person… . 

We have 365 days in the year normally, right (http://scienceworld.wolfram.com/astronomy/Year.html)?

Well, Adrian Dudek said it well: Had we 367 people then, we would guarantee that at least two of those had their birthdays coinciding.

Why 367 and not 368 in this case? Because there are years of 28th of February and there are years of 29th of February, therefore there are years of 365 and years of 366 days, this following the Gregorian Calendar, which is the one we have adopted in the United States, for instance (http://www.timeanddate.com/calendar/gregorian-calendar.html).

If we have more than 31 people, we should also get a coincidence in terms of day of the month... .

More than twelve, and we should get a coincidence in terms of month… .

That is the only way we can guarantee such a coincidence, however, despite the assertions of the famous mathematicians in national and international TV and the text of Adrian Dudek.

It may happen that we get the proclaimed 27 people a few times, say ten, and in all those times we get at least two birthdays coinciding precisely. We call that luck, just like with the game Lotto and the Monty Hall Problem. Notwithstanding, we have to stop trying to make what is accurate, scientific, become something that gets the attention of the so-called uninterested or common people: It is all right to try to be popular and to try to make everyone like Mathematics or appreciate it, but it is not all right destroying our Science, make us all look like idiots as a class (scientists) just to appear on a TV show. 

We have to, basically, stop with the six-degree-kinda thing (it is definitely not true that we are always at most six degrees away from each other. Please see our paper, Starants II (http://www.innovativejournal.in/index.php/ajcem/article/view/108), for some obvious argumentation).

____________________________________________________________

DO YOU WANT TO PLAY LOVE GAMES? GET THE BOOK HERE!

____________________________________________________________

    

The Monty Hall Problem and a few moments of shame for Modern Science and scientists: Newcastle, 2000, Australia

Dr. Graham Priest (http://www.st-andrews.ac.uk/philosophy/old/gp/gp.html) gave a talk at the Newcastle University (http://www.newcastle.edu.au/) in 2000 stating that the Monty Hall Problem (http://www.nytimes.com/2008/04/08/science/08monty.html) was a problem worth studying because it defied the laws of Mathematics.

Whoever was there heard.

Well, all that we know about this problem is that it appears to be connected to a TV show where a car would be hidden behind a door. The other doors would have something else behind them. The person participating in the game would choose one of the three doors and if they guessed right, like if they chose the door that had the car behind it, then they would get the car. The presenter would then open one of the two doors that remained (not chosen) after the person made the first choice. That door would never have the car. 




It is then said that they have studied the problem from the TV show and if the person changed their choice at that stage, like after the revelation of the contents of one of the two remaining doors, then the person would have more chances of getting the car because studying the history of the show they reached that conclusion, like, statistically, that would be the case.

A fellow from LinkedIn (http://www.linkedin.com/) came up with a poor draft on a possible explanation. Because it is not correct, we will not mention his page or name here.

He listed all possible first choices and then drew a map of possibilities, what we call tree, to prove that truth. 

The problem is that his map is incomplete.

Please notice that we have three choices available on the first time the question is asked and therefore one chance in three of finding the car (about 33% or 1/3). 



One of the doors is open and no car appears. 




We are now left with two doors. One of them has the car. 




Has the probability changed? Yes, sure! Now we have 50% or 1/2 of chance of getting the car, like it is more likely that we get it, since one door has been eliminated from the game and did not contain it. 




What happens if we change our choice? We then have 50% of chance again, is it not?

Notice that the first choice will never have any effect on the second because the presenter always opens a door without a car behind it. 




Notice that the door is not closed again and mixed, as we do with the cards game and we have not eliminated the possibility of getting a non-car by opening that door. 




If we considered those to be two sets of doors and we had to get car in both to win, then we could think of conditional probability, right? If we chose the car on the first choice, 1/3, and the car on the second choice, 1/2, then we would have a probability of 1/3 x 1/2 of getting it, that is, of 1/6 or app. 16.67%.

The way the problem is, however, we have about 33% of chance with the first choice. If we change the choice after the door is open, then we have 50% of chance of getting it and if we stick to the same door, we also have 50% of chance of getting it. 




Why is it that the history of choices of the show seem to conflict with this conclusion then? As in Brazil, with Silvio Santos, more than likely (same sort of game), we can only assume that people would change doors more rarely and it was then the choice of the production to swap the position of the car so that the person would lose, like that is just a TV show… . 

If we were to write a tree for this problem, it would look like this:

So, this is our complete tree of choices.

Basically, the first balls are the choice of the contestant on the first time and after that choice they will be presented with three doors again, but one will be eliminated, which is the one that appears with the cross.

With that, we have the following couples of results as our possibilities set:

{(D1,D1), (D1,D2),(D1,D1),(D1,D3),(D2,D2),(D2,D3),(D2,D1),(D2,D2),(D3,D1), (D3,D3), (D3,D2), (D3,D3)}

The cardinality of this set is 12. 

Notice that in six of the possibilities there is equality between the first and the second choice, what then gives 50% of the results as possible. If we swap, that is, if the second choice or the second member of the couple changes, then we have the other 50%.

We have then proven, also through tree of possibilities, that the chances are 50% for those who stick to the same door and 50% for those who change on the second opportunity.

We think that this settles the possible problem with the Mathematics involved.

Were the game honest, we probably should notice some similarity of wins between the people who swap and those who stick. Notice however that luck is luck and some numbers get out of the globe/bag more times than others with draws controlled by auditors in the popular games we have, say the game Lotto in Brazil. 




If we could draw a rule that were deterministic, this would be purely mathematical, not also statistical (probability falls to the side of Statistics). Whatever is statistical is not supposed to be deterministic by nature, so that the results are not predictable in terms of real life. The best we get is a percentage in terms of chances of getting it right.

Also, if we could predict results, then we would always win, what would obviously make profit impossible to the side of the organization running the game.

This problem perhaps exists to illustrate the differences between what is statistical and what is mathematical in the same way that The Sorites Paradox exists to illustrate the difference between purely human language and machine or mathematical language (Mathematics is based on Classical Logic, which was the first logical system used by machines. The first circuits we had were all based on Classical Logic, and that is why they had only two states: Open or closed. Basically, to have our air-conditioners, the modern ones, we had to use Fuzzy Logic, which is a logical system that accepts more nuances, not only two truth-values, let’s say).

______________________________________________________

DO YOU WANT TO PLAY LOVE GAMES? GET THE BOOK HERE!

______________________________________________________

    

Negative numbers and basic operations

It may look as if we are over it, but the negative numbers still represent a puzzle for many.

Ms. Ruth McNeill[1] found it very counter-intuitive to consider that a negative number times a negative number comes out to a positive number. According to the source we have consulted, this lady claimed that multiplying intensifies something, and thus two negative numbers multiplied together should properly produce a very negative result.

All in human knowledge is obviously convention, for even the most guaranteed of the findings, say a person is definitely with worms in their digestive system, and we should therefore remove those, if we see a roundworm coming out of their body, depends on a set of premises.

So we have a body?

We obviously have a body for as long as our senses tell us that there is a difference between us and the others that is determined by our skin, for instance.

Trivially, if we find out that our eye has a matrix that is not evolved enough, and an animal of some sort has a better matrix for their eye, and we decide to use their matrix, and, in doing that, we see no separation between our bodies and the bodies of others, then we will start believing that the worm is not inside of our digestive system: It is inside of our common body instead.

If things get worse, and our new eye matrix reveals that we are actually also connected to the land around us with no distinction, then the worm is another being fighting for its life in the only environment that is available to it.

In this case, our worm may become our next Tasmanian Devil, like it may need to be preserved, and perhaps it is then OK if a few pieces of us die in order for it to keep on existing.

If even physical things, which we take for granted, are actually products of our imagination, all founded on a set of premises, or assumptions, imagine the rest… .

Human knowledge is fragile.

Inside of the World of Mathematics, we have the negative numbers, which could be seen as the reflection, in the mirror, of the non-negative and non-null numbers.

We should be able to explain all basic mathematical operations with the negative and non-null numbers in the same way that we explain the basic mathematical operations with the non-negative and non-null numbers.

It could be possible that Ms. Ruth were right and all our conventions, those that have to do with the negative numbers, were actually not logical. If that were true, however, our explanations would have to be odd, unnatural or counter-intuitive, as she said they were.

When we add two non-negative and non-null numbers, we think of two sets of items being merged, then of us counting the items, and coming up with a figure in the end. Some  of us, however, think of the ruler and then imagine a particle moving from the starting point, the 0, over as many units as those we have in the first number, then over as many units as those we have in the second number, and finally stopping. Those would go to the ruler and take note of the number that is found under the particle as they finish doing that.

Well, if we put the ruler in front of a mirror, then we will see its reflection in it. If we now position the ruler in a way to have its zero connected to the surface of the mirror, we will see another ruler in the virtual world that we have just created.  As in Physics, we translate inside of the mirror into a '-' sign. Now, if we forget the '-' sign of the amounts involved and perform the requested mathematical operations in the way we described before, but look at the mirror, at the reflection of the result, we will have our actual result. We will then have both quantities together; therefore a value that represents that, but, this time, this value will appear to the right side of a minus sign.

Now, let’s think of our bank accounts. Let’s think of the minus sign as a synonym for debt. We would then increase our debt as we increase the quantity of numbers with negative signs in our sum, right? Because it would still be a debt, we would have to have the minus sign in front of the result of the sum.

If we had a debt of, say, two dollars, and we wanted to make all our seven friends owe as much as we do, therefore we would like to copy our situation for each one of them or we would like to multiply it, we would have minus fourteen dollars, since that would be two dollars in debt seven times or 7 x (-2).

If our friends then decided that in fact they owe that money but left their seats, and they were sitting at our table before, and they all have put the money they owed us over their seat before they left, we then would have -7x(-2)=14.

We could also think of the situation as being our seven friends who have paid their debts with us, therefore our 7 x (-(-2)), that is, our seven friends that now can have that debt taken away (with a minus) or 14 dollars entering our pocket since it is two for each.

Besides, if we have a positive balance in our bank account, say AU$ 4, and we then acquire a debt of AU$ 10, we know that it is missing AU$ 6 for us to be OK with our bank, that is, we have a -6 result.

If we divide a debt of, say, AU$ 10 by two people who do not exist, we are probably trying to lure the Taxation Office because we actually have a profit that we cannot explain in our result with our business, let’s say. That means that we actually have a non-negative balance to deal with. A 10-dollar debt that has never existed divided by two people that have never existed ((-10)/(-2)) can only mean an excess of 10 dollars, therefore a +10 or a 10. In this case, we actually have an excess of 5 for each person.

Another situation that could explain the process of division involving two negative quantities is that in which we have the quantity of people we had before owing us minus two. They all owed AU$ 10, therefore had a debt of AU$ 10 (-10 x 2). Now they have paid their debt, so that we have (-10)x(-2), therefore we have a non-negative 20 as an entry in our bank statement… .

On the other hand, we could have AU$10 to explain to the Taxation Office. We would then come up with two people who owed us money, but two people who do not exist, in order to even out all and not pay tax. We would then have a 10-dollar profit to divide by 2 people who do not exist (10/(-2)). Well, the result, in this case, can only be that each one owed us AU$5, therefore each one got a -5 for their account… .

If we had a debt of AU$ 10 to divide by two people that actually exist ((-10)/2), then we know that each one would be getting a debt of 5, that is, a -5.

It then seems to us that Ms. Ruth McNeill could be more accepting of the mathematical rules if she had access to our explanations.

If we could go back in the past, is it not? Only if we could… .

---

[1] Ruth McNeill, “A Reflection on When I Loved Math and How I Stopped.” Journal of Mathematical Behavior, vol. 7 (1988) pp. 45-50, as mentioned at http://homepages.math.uic.edu/~saunders/MTHT591_F12_crypto/supplements/4gelfand.pdf (Bonnie Saunders, September 2012, as seen by us on the 10^th of July of 2013)

____________________________________________________________

DO YOU WANT TO PLAY LOVE GAMES? GET THE BOOK HERE!

____________________________________________________________