Wednesday, April 13, 2016

124. Infinite Monkeys

What's the probability that the Philippines would ever have a good President? There are six candidates this year. There will be another elections on 2022, assuming the winner would not abolish the Constitution. But either way, it is not a guarantee that there would  be elections in 2022 or that the elections would be the avenue for having a good President. There are many modes for having a good president -- election, coup d'etat, revolution, succession, etc. --- and in each mode there are more permutations,  so the probabilities also grow exponentially. If, however, we consider that the question is posed with the underlying assumption that this is a question for all eternity, that is -- what is the probability that Philippines will ever have a good President at some point in history given an infinite number of modes within the infinite history of this country? The answer is just about the same chance that 110 million monkeys, or as many as we want, typing on a keyboard would produce the words "Banana Republic" at some point given an infinite number of keystrokes. This is the Infinite Monkey theorem, the standard solution of which goes like this,

Let An be the event that the 
nth monkeys type Banana Republic. Then,  if there are mcharacters on the keyboard and Ncharacters in Banana Republic , PAnmN for each n. Furthermore, the An are mutually independent. Hence, by the second Borel-Cantelli Lemma, since 
n1PAnn1mN
infinitely many of the events An occur i.e.infinitely many monkeys will type Banana Republic. Thus, there is a single, unitary, solitary, lonely, individual, isa, usa, uno, jeden, man, nomes, or 1 chance in all eternity that the infinite monkeys will type Banana Republic as there is the same chance that the Philippines would have a good President. Of course, as a pure thought experiment where at least one factor is infinity, there is always one chance the randomness will produce the desired result.  Yet, what would be the answer if we apply this theory to practical reality, and if we ask, given the permutations of the number of candidates and available modes of having one, what is the chance 110 million Filipinos would have a good President in 2016? Well, the University of Plymouth once had six monkeys in a room with six computers to test if the monkeys could type anything remotely Shakespearian. After a month, the primates could produce only five pages of typewritten text consisting mostly of the letter "S". That is, after the monkeys found out there was another use for the keyboard that did not involve urinating or defecating on the machines. Yet, it must be emphasized that the Philippine voters are not just a random machine or monkeys holed up in a room with a computer. There is an active intent to search, find, and vote for a good President. Thus, the chance of having a good President in 2016 should be greater than one. That is, of course, if the majority of voters would find better use for their ballots that did not involve defecating or urinating on them literally and figuratively.

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