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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I think I don’t really know what I mean when I say that “Context Matters”…yet.

Context matters sounds like a banal platitude, a cliché, doesn’t it? Well, it might be…. but… as we are reminded by David Foster Wallace in his This is Water speech, sometimes a banal platitude hides the most important truths, truths that become invisible to us. These truths are so present that we forget about them, just like when you’re wearing your  most comfortable clothes.

Daniel Kahneman writes in an article published in December 2003 in the journal The American Economic Review:

Perception is reference-dependent: the perceived attributes of a focal stimulus reflect the contrast between that stimulus and a context of prior and concurrent stimuli.

I stopped reading that article right at that point. A question popped into my mind that I thought required a little bit more than a dictionary-lookup and a Wikipedia search…

What is the difference between reference-dependent and context-dependent?

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