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).

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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).

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