Probability of Typing a SonnetDate: 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/ |
Search the Dr. Math Library: |
[Privacy Policy] [Terms of Use]
Ask Dr. Math^{TM}
© 1994-2015 The Math Forum
http://mathforum.org/dr.math/