Probability of Typing a Sonnet

Date: 9/2/96 at 10:35:32
From: Durand Sinclair
Subject: Probability of typing a sonnet

Hi,

I wonder if you can help me. I've just heard an argument for the 
existence of God that used statistics. As I am a little unsure of my 
stats, I was wondering if you could read it for me and make sure that 
the maths side is correct, and where, if at any place, the argument 
breaks down. I am not trying to proselytize here. I am trying to work 
this stuff out for myself. But I don't want to be bamboozled into 
accepting this argument without double checking it with someone who 
really knows their Maths.

COULD A MONKEY TYPE A SONNET?

When talking about the chances that life could have evolved by pure 
chance, there's a famous analogy of a monkey typing away at a 
typewriter. Most of what is typed is utter rubbish, but occasionally, 
by pure chance, it'll type one of Shakespeare's sonnets.  This analogy 
falls down when you actually calculate the probability of typing a 
sonnet out. 

If we were to take one at random, the one that starts "Shall I compare 
thee to a Summer's day?" (which is the only one I studied in high 
school), and look at it, we notice that it has 488 letters in it. It's 
not a particularly long or short sonnet. All of Shakespeare's sonnets 
have 14 lines, and so would have around that number of letters in it.  
So what are the chances of typing out this sonnet by chance? Well, 
we'll ignore the punctuation marks, space bars and enter keys. The 
chances of typing out any particular letter is one chance in 26. The 
chances of typing out two particular letters are 1 in 26 times 26 (ie 
26 to the power of 2).  The chances of typing out all 488 letters at 
random are one chance in 26 to the power of 488. To make it easier to 
visualise, that's 1 in 10 to the power of 690.

Sounds fine. So the next question is, how much time has there been in 
the universe, and has there been enough time for our monkey to type 
the sonnet.  Well, according to the physicists, no. The accepted view 
is that there was a Big Bang that started the universe off some 10-20 
billion years ago. This translates as only 10 to the power of 18 
seconds since the beginning of time. Add to that, science also 
believes that the universe has 10 to the power of 53 atoms in it. 

NOW FOR THE CONTROVERSIAL BIT --

Therefore, if since the beginning of time, every atom had been typing 
a sonnet every second, it would still be unlikely that the sonnet 
would have been typed. If we multiply the number of atoms by the 
number of seconds, we will get the number of sonnets that have been 
written so far. (And because we're multiplying powers of ten, we have 
to add the indices.) So far then, our atoms will have written out 10 
to the 71 sonnets. (53+18=71) As a fraction of 10 to the 690, this is 
1/(10 to the 639). Which means that the probability of the sonnet 
being typed, given the best possible scenario in the universe, is much 
much more improbable than finding a particular electron you lost 
somewhere in the universe. 

In short, there is no way that ANYONE should believe that there is 
even the REMOTEST possibility that a sonnet could ever come about by 
pure chance. Not even if every atom in the universe was able to type 
one sonnet a second since the beginning of time. And here's the other 
thing. A sonnet is nowhere near as complex as a cell.  So that 
couldn't have happened by chance either.

Okay. That's the argument. Its been troubling me greatly since I heard 
it, and I would GREATLY appreciate it if you could check the stats 
side of the argument to make sure a) it's sound, and b) the stats do 
in fact point to the conclusion that there would have to be more than 
pure chance at work to write a sonnet in the given amount of time in 
the universe. Also, do you know of any "Ask an expert" pages that deal 
with molecular biology, or someone who could tell me enough about how 
a cell works so that I can judge if it really is more complex than a 
sonnet? 

Thanks very much. I really do appreciate it :-)

Durand Sinclair
(from Sydney, Australia)

PS: I've read a couple of physics books to see what the smallest 
amount of time would be. (Perhaps a second is too great a time.) One 
book I read said that the smallest amount of time a physicist 
considers is an instant, which is 10 to the -43 of a second. It still 
seems not enough time. The view of 10 to the 18 seconds since the Big 
Bang is pretty standard. They calculate it because astronomers have 
noticed that the universe is expanding, ie the stars are getting 
further and furhter away from each other. So they extrapolate back, 
and find that between 10 and 20 billion years ago the Universe must 
have been at a single point. The idea that there has been lots of 
explosions, with the universe expanding then contracting forever 
(which would give us as much time as we want) is generally discredited 
now. This is the only Big Bang that's been. It's a long explanation as 
to why.

Date: 9/2/96 at 13:1:39
From: Doctor Jodi
Subject: Re: Probability of typing a sonnet

Hi Durand! I can comment on a few things:

There is an ASK A SCIENTIST page at 

  http://medinfo.wustl.edu/~ysp/MSN    

which may help on the molecular biology and cosmology side.  
Cosmologists, astrophysicists, etc. are FAR from coming to a 
conclusive answer about the age of the universe or its nature 
(oscillating or non-oscillating, finite or infinite, etc.)  Their 
conclusions might affect your conclusions greatly.

The size of the universe is also not definitely known, so I'm not sure 
how to judge the second half of the probability calculation. However, 
the probability of writing a sonnet is correct if you mean that typing 
out all letters CONSECUTIVELY, etc: "The chances of typing out all 
488 letters at random are one chance in 26 to the power of 488."  You 
should know, though, that probability is an inexact science which 
relies heavily on averages and large samples.  

I hope that I've helped you a bit. Another Doctor will comment further 
on the probability and fuzzy logic aspects of your question.

-Doctor Jodi,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 9/3/96 at 20:26:12
From: Doctor Tom
Subject: Re: Probability of typing a sonnet

Hi Durand,

The probability calculations above are roughly right, but the 
conclusions drawn from them are outlandish.

This calculation, for example, calculates the probability of a monkey 
typing A PARTICULAR sonnet.  How many possible sonnets are there?  
Does it have to be in English?  Why? ...

Let me use a similar argument to show that you can't possibly exist.

Your DNA is composed from 23 chromosomes from your mother and 23 from 
your father.  But as those chromosomes were being formed for the egg 
and sperm cells, a bunch (perhaps 5, on average) cross-overs occurred, 
more or less at random along each of the chromosomes, so for each 
parent there were roughly 5x23 = 115 more-or-less arbitrary selections 
chosen from the half-million proteins your DNA codes for. 

This ignores the fact that cross-overs can occur within the proteins - 
I'll do the conservative case.  How many ways can you choose 
115 crossovers from a half-million cross-over spots?  It's roughly 
500000^115, right?  And that's just for your father; there are similar 
probabilities for your mother.  And so the odds against your 
particular DNA configuration are MUCH smaller than 500000^115. This is 
a very conservative guess, but it has a probability of FAR less than 
the probability in the monkey/sonnet example above.

So clearly, it's overwhelmingly unlikely that you exist!

What's wrong, obviously, is that almost any combination of cross-overs
will make a viable human.  If, before you were conceived, someone had 
wanted to calculate the probability of getting EXACTLY you, the odds 
were mind-numbingly against it.  But if someone simply asked what the 
probability is that there would be some viable child, that probability 
is very high.

The point is, there are a gazillion POSSIBLE different humans - and
any one particular one is incredibly unlikely.  But the odds are 
overwhelmingly likely that SOME human will result.

Similarly, there are a gazillion possible life forms, and any of them 
"works".  Consider how many organisms have lived on earth since the 
beginning - trillions of trillions of bacteria, reproducing every 20 
minutes for 4 billion years.  I'd be willing to bet that they're 
almost all slightly different - different counts of water molecules, 
et cetera.  All are possible "solutions" to the problem of "typing" a 
living organism at random.  Do you think what we've seen exhausts the 
possibilities?  Not even close.

The other terrible error is that arrangements of atoms were not tried 
at random until a working cell appeared. All you need is some 
combination that catalyzes reactions that make a similar molecule 
every now and then, or even catalyzes a loop that eventually makes a 
similar molecule every now and then. These are selected for. The 
combinations that work better will get more common.  Better 
combinations will replace worse ones. That's the whole idea of 
evolution.

There are 60-atom molecules that can reproduce themselves.  Do you 
think that's the ONLY such molecule?  No - there are probably 
trillions of such things. And then there are the 61-atom 
possibilities, and the 62-atom possibilities, and so on.

But don't bother to argue with whoever told you the example - trust
me, it won't do any good.  They "know" the answer already, and I'm
sure mere facts won't alter their reasoning :^)

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 9/5/96 at 9:26:57
From: Durand Sinclair
Subject: Re: Probability of typing a sonnet

Hi Dr Tom :)

Thank you for writing back to me about the probability of typing a 
sonnet. I'm very glad that I wrote to you, because I had only heard 
one side of the story. Without your Web page, I would probably not 
have heard a different viewpoint. 

I had no idea that all you needed to start with was a molecule as 
simple as 60 carbon atoms! But thinking about it, it leads to the 
question that if its just a purely mechanical process of one Carbon 60 
atom producing another, why would any one produce something that ISN'T 
a Carbon 60 atom? For evolution to occur, we would need something that 
produces some kind of offspring that is just a tiny bit different from 
the parent so you can get freaks that start a new species. So you need 
(at least) two bits of "genetic code" (even for an inanimate thing 
like carbon 60), otherwise it wouldn't be able to manufacture slightly 
different offspring. 

But the main thing that the other argument missed is indeed that who 
says we should look at that PARTICULAR sonnet or living cell. That 
point you made gives a lot more chances for some sort of life to 
happen randomly. 

I speculated on another answer after I sent my other email: 
ITS NOT REALLY CHANCE - The idea of "atoms combining by chance" 
conjures up images of atoms that really don't care who they combine 
with or what combinations they go for. But that's not true. Atoms 
prefer to combine with particular other atoms, and will "choose" one 
over the other. (e.g. when you mix HCl with Sodium hydroxide (NA O H) 
you get salt and water because each one prefers to be in the new 
state. (Sorry if I'm being anthropomorphic ;)  Anyway, perhaps there's 
some characteristic about carbon that it likes to form chains. Or to 
put it more scientifically, it reaches a more stable equilibrium that 
way. So its not PURE chance that causes carbon to form into living 
cells, its directed by its own electrical forces (or whatever it is 
that causes one atom to be attracted to the next.) This would mean 
there's a much greater chance of living matter happening than just 
"randomness". Then again, if its that simple to get life forms, we 
should find that new life forms are popping up all the time. I don't 
know, maybe they are. Do you know?

More usefully, do you know of any good books on this sort of subject? 
I have finished High School, but did not study science at university. 
If there are any books that deal with evolution, statistics, the 
origin of life etc that you know of that are readable, could you let 
me know? Especially one that explains the "crossover in protein cells" 
example you wrote me. (I'm still a little unsure of what a crossover 
is.)

Thanks :)
Durand Sinclair

Date: 9/11/96 at 20:33:33
From: Doctor Tom
Subject: Re: Probability of typing a sonnet

Hi Durand,

Sorry it took a few days to get back to you.  I am a sort of
"volunteer Dr. Math", and I check in whenever I have time.  I have
a real job too, unfortunately. :^)

But to answer your questions --

Any kind of chemical reaction will never be 100 percent accurate - if 
the atoms get hit by cosmic rays, things can be knocked apart and 
they'll reform in a possibly different configuration.  In other words, 
the BEST you can hope for is a highly accurate (but not perfect) 
reproduction rate.  There will always be some "mutations".  Our cells,
in fact, have elaborate mechanisms to check for accuracy in copying,
but there are still occasional mutations.

I'm sure that early life had lots of mutations, but (since at the
beginning, there was virtually no competition at all), any molecule
that could OCCASIONALLY make a copy of itself exactly had a HUGE
"advantage" over molecules that never could.  "Mutations" of this
molecule that "reproduced" more accurately would have an advantage 
over the original, and would tend to become even more common, and
so on.

In fact, you don't even need a molecule to make a copy of itself --
it just has to make something that makes something that makes the
original -- any sized loop will do!

Under these conditions, it's actually hard (for me) to imagine that
life wouldn't evolve!

Now the reason that life isn't evolving all the time isn't hard to
explain either.  Imagine a very badly reproducing molecule that arises 
in the oceans today.  It will find itself instantly in competition 
with the most sophisticated biological machines that evolution has 
been able to create in 4 billion years.  What do you think its chances 
are?  Right... just about zero.

There may have been amazing "battles" of this sort very early on, but
once one form wins out, each new life form that tries to get a toehold
will face tougher and tougher competition, and today, it's basically
hopeless (on Earth, that is -- I suppose you've read about the 
possible discovery of life in the Martian meteorite).

To learn about cross-over (in a chromosome, not a protein), I'd just
get hold of a modern freshman college biology textbook.  If possible,
get a text that takes a more biochemical view, although I think that
these days, most do.  I just went into the local college bookstore to 
see what the freshmen were using in Biology 1 and got that!

But there are plenty of books written for the layman that are great.

I highly recommend "The Selfish Gene", by Richard Dawkins.  It made a 
lot of evolutionary biology crystal-clear to me.

Also highly recommended is "The Red Queen", by Matt (or Mark, I 
forget) Ridley.  It provides great explanations of competition, 
evolution, et cetera.

A very tough read (not because it's highly technical - just because
the information is so dense) is "Darwin's Dangerous Idea", by Daniel 
Dennett.  He talks not only about the application of evolution to 
life, but also to many other things, including the generation of
life.

Many biologists consider "evolution" to concern itself only with life
after it somehow formed, and call the process of the creation of life
from non-life "abiogenesis".  Needless to say, with this definition,
"evolution" is far better understood than "abiogenesis".

Off the top of my head, I don't know any great books on abiogenesis.
I know that Carl Sagan is highly interested in the topic, and you 
might look at some of his books for a lead.

Another great writer about biology for popular readers is Stephen
J. Gould.  He writes regular columns in the magazine "Natural 
History", and they are regularly collected in books.  All his books 
are great.  He is highly interested in paleontology and evolution, and 
you'll get some great insights from his books.  The other great thing 
about those books is that they're all composed of small (10 or 12 
page) self-contained essays, and if you find you're not interested in 
one, you can skip to the next with no penalty.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/