The Borel-Cantelli Theorem

Let \{A_j\}_{j=1}^\infty be an infinite sequence of events.

If the A_j are mutually independent events and \sum_{j=1}^\infty P(A_j) diverges, then P(\{ A_n {i.o.} \} ) = 1. Where \{ A_n{i.o} \} reads “A_n infinitely often”.

So, what does it mean? Well, we can think of it like this:

Suppose you have your brand new email account, and you are thinking about a really safe password, one that is “unbreakable”. Well, the above theorem says that no matter how super long and complicated you make your password, eventually it can be “broken”, so it is rather a question of asking “How long will it take for someone to break my password?” instead.

Another way to think about it:

You have a coin, and you are going to start flipping that coin until you get Tails, Tails, Head, in that order, consecutively, three times. You will eventually get it, it’s just a matter of when, not if.

There are many other examples that we can come up with, but I suggest that you come up with some of them. See you next time!

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What’s the deal with zero-ones and turkeys?

I am really excited. Today I found a mathematical theorem closely related to Nassim Taleb’s Turkey Problem . In this post I’ll just retell the turkey problem as stated by Taleb, the next post will include the mathematical theorem that I just found today (so you can give me a couple of minutes to digest this math concept); finally, the third post will include the link between them. Ok, here we go.

The Turkey Problem

Imagine you are a turkey, in a really nice farm. Everything is awesome. Today is January second, and you just woke up in this great farm – you don’t remember where you were before, but that’s ok.

Time goes by, let’s say a week, and you notice that other animals, like, say, the pigs, get taken away by these two legged creatures only to never be seen again. A couple of months go by, and every week, you see them take away at least one pig, and it never comes back. Something bad must happen to them, you assume. Now, eight months have gone by, from your previous observations you come up with a pretty clever theory (according to your turkey brain):

Careful observation indicates that the two legged creatures take away pigs, and only pigs, so turkeys, such as yourself, never get taken away by the two legged animals.

Then, as you clever reader might have imagined, November comes…

If you are still curious and can’t wait for the next post to know what’s going on, the title contains a huge hint on what you should be looking for… See you next time!

Forms in Nature

The 1 0 trap

It makes you think there is or isn’t something there.

Writing as a freeing exercise prior to a technical phone interview

“El Dinosaurio” by Augusto Monterroso

Cuando despertó, el dinosaurio todavía estaba allí.

That, right up there in italics, is the world’s shortest story. 

When one thinks of writing a story, a short story, or any story whatsoever, we probably do not think of the above.

If we were to write those words, in that order, all of a sudden, we probably would just consider it the first part of something, a sentence that is neither here nor there, maybe even a possible ending. But, a full story? In just ~66 characters? 

Now, I would like you to consider this: Was Monterroso able to create this short story, a complete short story in 10 words,  because the cultural context had developed in such a way that it was accepted by its peers as a short story? Had this story been written by, say, Shakespeare, do you think it would have counted as a short story? What if it had been written by an old Babylonian priest? (Assuming that a Babylonian priest would know what a dinosaur is/was or maybe substituting the word dinosaur by lizard or alligator or some other animal known to such hypothetical priest)

My own, personal, answer is… I don’t know.

One possible way to view reality is via semiotics: the current state of affairs that we call the world is composed of a series of symbols whose meanings change constantly, via paradigm shifts – I am totally abusing Kuhn’s original interpretation – the old symbols being replaced by new ones, and probably becoming a mystery to us, in other words the whole signifier/signified/sign/symbol thing.

I can always play around with the idea that I understand what Leibniz, or Mercator for that matter, meant by infinitesimals, but my mental constructs are already “infected” with other parasites that make the – original – concept of infinitesimals nothing more than philosophical ramblings. I cannot change this. (See also limit)

 Now, I am not saying that it is not possible, all I am saying is that through my experience I have found that it is impossible for me to exactly know what was meant by infinitesimal in Leibniz’s time. What I think I am trying to say is that his contemporaries might have had a clearer grasp of what the concept meant for Leibniz himself than we will ever have. The accumulation of – more accurate, more precise – concepts via science – and, by all means, science is awesome – has created a context that permeates everything else. No way to go back really.

 Can you think of  a way?

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