Flipping Our Coin

Flipping our Coin
There’s a little bit of “math” in this post, but please bare with me…I really do have a point.

 

The probability that a fair coin, when flipped, will turn up heads is 50% (0.5, ½, ‘1 in 2’, however you’d like to phrase it). Even a child can tell you that.

 

Now, what is the chance that the same coin will come up heads twice in a row (by that I mean, what are the chances before the first throw)? Any high school math student can tell you that it’s ½ times ½, or ¼. The basic rule is simple to remember: the probability of any two independent sequential events both happening is the product of the probability of both; so it’s (1/n)x, where n is the number of possible outcomes (2, in the case of the coin) and x is the number of independent events (how many times we flipped the coin). So, for two coin tosses, it’s ½ x ½ = ¼, for three tosses it’s ½ x ½ x ½ = 1/8, and so forth. Of course, this only applies before you toss the first coin, since any given throw has a ½ chance of being heads; even if you’ve tossed 10 heads in a row, the chance of the next throw being heads is still ½. I’m quite sure that most people have no problem conceptualizing this, so let’s make things a little more complex.

 

If I flipped ten straight heads, you might think something was a tad askew. What are the chances of getting ten straight heads, you might say? Well, just apply the same logic as before: (1/2)10 = 1/1024. Put into words, if we flip a coin 10 times (let’s call this one ‘set’), we would only expect to have a ‘set’ end up with 10 heads in a row once for every 1024 sets. Make sure you grasp this before moving on.

 

Now, a 1 in 1024 chance of getting 10 heads in a row seems pretty unlikely. It appears far more likely that I’ve biased the coin in some way. But let’s say that you picked the coin and did the flipping, and you got this result. You’d be shocked, wouldn’t you? In fact, let’s assume you flipped the coin 100 times and got 100 straight heads. You’d be blown away! The chances of that happening are: (1/2)100 = 1/7.9 x 1031. In words, this means one chance in 79 million million million million million (that’s 79 with 30 zeros after it). By comparison, the universe is 13.9 billion years old (more on that later), in which time only about 1017 seconds (1 with 17 zeros after it) have elapsed. Suffice to say, 100 straight heads is pretty unlikely. If you got 100 heads, you would be absolutely astonished, and convinced that the coin was “fixed”.

 

Now, flip the coin 100 times. The odds are very good, of course, that you’ll get something other than 100 heads. But wait, what are the chances of getting any particular result (that is, any given pattern of 100 heads or tails)? Well, apply the basic probability theory: the chance is ½ for each flip, meaning that the probability of having any particular pattern is (1/2)100, which is the same as the probability of getting all heads.

 

We’ve already agreed that if you had seen all heads, you would have been astounded by the result. And yet, the pattern you did get inspired nothing but the boredom that comes from being forced to flip a coin 100 times. Why should this be?

 

The reason you are amazed by 100 heads (or tails) and not any other, equally likely pattern is that you see a recognizable pattern in the 100 heads; everything else just looks random to you, so you ignore it. Human beings are spectacularly good at seeing patterns. Unfortunately, we are also spectacularly bad at interpreting them, so we don’t think twice about the seemingly random patterns.

 

There are numerous examples of this that occur throughout your life. If you won the lottery, you’d be amazed at your dumb luck (your odds, in a typical “6 from 49” game are 1/13,983,816; the math for this can be found at http://en.wikipedia.org/wiki/Lottery_mathematics). But if someone else wins, you’re not surprised at all (they will be, though, and this will become important later). You inherently accept that as long as enough people are playing the lottery, someone will win it, but that the likelihood that it will be you is very small. This is precisely analogous to the 100 heads example: the chances of getting 100 heads (analogous to you winning the lottery) is very small, but the chances of getting some combination of heads and tails is 100% (analogous to anyone but you winning the lottery).

You can carry the analogy even further, if you’d like. Let’s say we bet on the outcome of the coin flips. For the sake of practicality, let’s lower the number of flips to something more manageable; 24. (1/2)24 is equal to 1/16777216. Now, let’s gather 16777215 of our closest friends, and have each one “bet” on one unique combination of heads and tails; I’ll take 24 heads. What are the odds that I’ll “win” the bet (i.e. we get 24 heads in a row)? Simple: 1/16777216. What are the odds that any other given person will win? 1/16777216. But someone will win, and that person will be shocked to have won. They might even think the game was rigged for them to win. But why should anyone think this? Someone has to win the game.

 

I share this little example not to highlight my love of coin flipping, or lotteries, or even of probability theory, but rather to highlight how bad our conception of “chance” really is. I cannot tell you how many times I’ve heard something along these lines:

 

“Life is too complex to have arisen by chance. It must have been created by an ‘intelligent’ designer.”

 

What nonsense.

 

What are the conceptual “odds” of producing, de novo, life as complex as ours (notwithstanding the fact that “complexity” is a relative, rather than absolute, quantity)? (EDIT: I want to clarify that ‘life’ as we know it is, of course, the product of random mutation and the non-random process of natural selection; in this post, I’m referring more to the conditions that allow for life, rather than the life itself. A small, but relevant clarification, I think) Let’s say, for the sake of argument, that the odds are 1 chance in 10200. 10200 is an enormous number; far greater than the number of milliseconds since the dawn of the universe, far greater than the total number of grains of sand that could fit into a sphere the size of our Sun. Huge.

 

Now, imagine yourself sitting on the sidelines just before the instant the first life came into being, and assume whatever happens next is totally random. What are the chances that life will develop into what we know today? Well, we’ve already arbitrarily defined it at 1 chance in 10200, so let’s stick with that. That implies that there is a far greater likelihood of some other outcome.

 

But…remember that there is a 100% likelihood of some outcome. Something has to happen. But each one of those “somethings” is just as likely as any other. Have you figured out where I’m going with this yet?

 

The person who wins our “coin toss” lottery is always shocked that they won, but the 16777215 other people who played are not shocked in the least. If the flips had come up slightly differently, someone else would have won, and they would have been shocked by it.

 

Similarly, the outside observer is never shocked by the outcome. To them, every outcome looks the same.

 

Why, oh why, should we be shocked that life is the way it is, and not some other way? The answer comes from our own arrogance (meant in the least pejorative way possible); we look for patterns. We are the winners of the universe’s “coin flip” lottery, and we are shocked by it. But if things had happened just slightly differently, we would have lost, and something else would have taken our place and had to deal with the same shock.

 

 

We can all intrinsically accept this when we’re talking about a lottery or a coin toss, but for some reason we have to ascribe some special meaning to our own existence, as though it were outside of what we intrinsically accept about probability; somehow, we think of ourselves as special.

 

 

This is not to say that we are not special; perhaps we are. But reliance upon chance to “prove” it is very poor form indeed.

I remain,

Michael

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27 Responses to Flipping Our Coin

  1. [...] I’ve talked at length about the lunacy of the claim that the apparent coincidence of our existence suggests that life was “designed” by god (to put a name to the designer); this argument is intellectually bankrupt and easily refuted by anyone with even a passing understanding of the Anthropic Principle. In fact, most people make the argument every day without knowing it. But when it comes to their own existence, which is too special to be left to coincidence, all reason and intelligence is abandoned in favour of dogma and Iron Age wisdom. [...]

  2. Bozo the Clown says:

    In the immortal words of Jim Carey in “Dumb and Dumber”: “so you’re telling me there’s a chance…” I readily admit to not having “even a passing understanding” of the Anthropic Principle. Even if I assume your “chance” argument is correct (which, due to the weakness of your argument in demonstrating either correlation or causality, I’m forced to do in order to counter), the question of the material universe remains. I.e. there is material in the universe – how did it get here? There is a universe – how did it get here? Don’t take me back to the “Big Bang” – all of the infinity wrapped up in that one small point had to get there as well – how? Chance can’t answer that, because even the chance ordering of any given thing has to assume the presence of that given thing. If that assumption cannot be explained then your argument is moot.

  3. edge100 says:

    the immortal words of Jim Carey in “Dumb and Dumber”: “so you’re telling me there’s a chance…” I readily admit to not having “even a passing understanding” of the Anthropic Principle. Even if I assume your “chance” argument is correct (which, due to the weakness of your argument in demonstrating either correlation or causality, I’m forced to do in order to counter), the question of the material universe remains. I.e. there is material in the universe – how did it get here? There is a universe – how did it get here? Don’t take me back to the “Big Bang” – all of the infinity wrapped up in that one small point had to get there as well – how? Chance can’t answer that, because even the chance ordering of any given thing has to assume the presence of that given thing. If that assumption cannot be explained then your argument is moot.

    Bozo,

    You might be surprised to learn that I don’t entirely disagree with what you’re saying (more on that in a second). However, I’m going to critique it anyway.

    Even if I assume your “chance” argument is correct (which, due to the weakness of your argument in demonstrating either correlation or causality, I’m forced to do in order to counter), the question of the material universe remains. I.e. there is material in the universe – how did it get here? There is a universe – how did it get here?

    A valid question. My post only deals with events after the universe came into existence. The question of “where did it all come from?” is beyond the scope of this post. However, three points:

    1. You cannot make the logical jump from “where did it all come from?” to “Jesus was born of a virgin”. Not saying you are, but that’s exactly what many people do (hence my post).

    2. Perhaps the universe was created by a larger power; I have no issue with that. There’s no evidence that this larger power is currently at work in this universe, however. This is deism, and most atheists have no issue with it.

    3. If you accept point 2, where did this larger power come from, or has it always existed? If it has always existed, why can we not say the same about the material universe?

    Chance can’t answer that, because even the chance ordering of any given thing has to assume the presence of that given thing.

    Look around you. The universe exists. It came into existence, as we know it, about 17 billion years ago. My argument is not moot, because the universe exists, and I have described, to the best of my ability, some of the laws of probability that would shape its development over time. My post was merely to show that stating that something is “unlikely” is meaningless; given a start, there has to be a winner, and the winner will always be shocked that they won (so much so that they’ll have to invent ‘cheats’ to explain it).

    Again, I have no issue with the concept of a “higher” power that created the contents of the universe (which would have been pure energy at the Big Bang) and the rules that govern it. What I take issue with is using this to claim that Earth and humanity have some sort of privileged place in the grand scheme. There is no evidence for that. I also take issue with turning this deist “creator” into some kind of personal god; that is a massive, unjustified leap (again, without some evidence).

  4. JR says:

    I think your post is excellent. And this is coming from a Christian. How do we know who’s right? I can’t prove my view through science. And (forgive my religious point of view here) I’m not meant to. If I ever convince you that Christianity is real it’s not going to be because I can refute the Big Bang or any other theory. That’s what a lot of “Christians” fail to realize.

    Anyways, I thought your post was interesting, so I made a spreadsheet with 1024 rows and 10 columns of random numbers. Then I totaled each row and added a max field at the bottom. Most of the time the max field had a 10, meaning that at least 1 of my 1024 rows had 10 1s. I didn’t really average it, but it was more than 50% of the time.

  5. [...] generated when a mob spawns (in other words, when you walk into an instance), it’s still possible. Even a flipped coin can come up heads 100 times in a row.So yes, even though we don’t think about it much, it’s totally possible to get [...]

  6. [...] generated when a mob spawns (in other words, when you walk into an instance), it’s still possible. Even a flipped coin can come up heads 100 times in a row.So yes, even though we don’t think about it much, it’s totally possible to get [...]

  7. Lesser_Immortal says:

    Please explain to me how life was able to defy the laws of physics (The laws of entropy specifically) for millions of years to go from a chaotic, maximum entropy state of primordial gases to a state of high information contained in DNA.

    The same goes for the universe in general. How was our universe able to go from a state of maximum entropy into the highly ordered state we see it in now, where stars form and planets arise. Wouldn’t the state of maximum entoropy simply have been a large, homogeneous cloud spread through out space?

    I believe in the laws of thermodynamics, not in a theory of evolution or a big bang.

  8. edge100 says:

    Please explain to me how life was able to defy the laws of physics (The laws of entropy specifically) for millions of years to go from a chaotic, maximum entropy state of primordial gases to a state of high information contained in DNA.

    First, you’ll need to explain to me your concept of the “laws of entropy”. In particular, explain the following equation:

    G = H − TS

    Once you understand the basic concept that the spontaneity of a process is determined by two factors (only one of which is entropy), you’ll be on your way. You’ll also have a good idea about how planes can fly, despite the massive decrease in entropy associated with flight.

    How was our universe able to go from a state of maximum entropy into the highly ordered state we see it in now, where stars form and planets arise. Wouldn’t the state of maximum entoropy simply have been a large, homogeneous cloud spread through out space?

    Once again, do not confuse entropy with free energy. Despite the fact that I don’t generally like to spend too much time doing research for creationists, I’ve done you the favour of going to wikipedia and getting this for you:

    http://en.wikipedia.org/wiki/Entropy_and_life#Gibbs_free_energy

    Read up, and learn.

    I believe in the laws of thermodynamics, not in a theory of evolution or a big bang.

    Luckily, you can have both; neither evolution nor the Big Bang are contrary to the 2nd Law of Thermodynamics (at least not the real law, rather than your creationist understanding of it).

    There is ample evidence for both evolution and the Big Bang to label both phenomena as “fact”. If you’d like a bit of the absolutely overwhelming evidence for common descent (i.e. evolution), you can go to:

    http://talkorigins.org/

    and specifically:

    http://talkorigins.org/faqs/comdesc/

    Take a few weeks to read the latter document (it will take you that long, even if you are a fully trained evolutionary biologist), then come back and discuss things like an intelligent person.

  9. elfy says:

    Goddamit, your post is kicking awesome. Now I’m confident I’m not the only one thinking that way

  10. roscoe says:

    Nice post.

    Very true and I’m Catholic. The problem with a lot of people, both religious and non, is that they aren’t educated. That’s why you get people like Lesser Immortal and people who would like to throw out all of Catholicisms morals and teachings because of one little thing.

    Plus, most people don’t know what the balls they are talking about when they refer to God; Atheists and Theists alike.

  11. edge100 says:

    Plus, most people don’t know what the balls they are talking about when they refer to God; Atheists and Theists alike.

    The problem is, who does know what they are talking about when they talk about god, and how do we know?

    What I mean is, as a scientist, I know about science (or at least a small part of it) because I’ve been trained in it. I can understand a similar training in the history of religious thought, or comparative religions, or what have you, but how exactly does one “know” about what god really is? Seems to me a bit like “knowing” what the Starship Enterprise really is; you can make up all the specifications you want, but at the end of the day, it’s only a TV show…

  12. roscoe says:

    Right. Who or What “God” is should be a purely personal question. Everything else should derive from reasoned thought, particularly morals (though, here, there is an interesting debate about whether morals can “exist” without “God” at all, but if youw ant to talk about that, it is for another discussion). You say you are an atheist, but I’m an atheist too if by “God” you mean some invisible personification with robes and a beard that plays chess and we are the chesspieces.

    However, God can also mean perfection, can mean the laws of nature. God can be a limit, as a purely mathematical perspective. My mathematician buddy and I always talk about how “God” can be analogous to the limit of the infinite convergent sequence 1/n. God, in this case, is 0 and we humans (the 1/n’s) can never actually reach God or “know” God. God could be love. I guess it’s more about having faith, faith simpliciter. It’s hard to explain, because I’m not asking you to have faith that abortion is wrong, that’s stupid, we should discuss that. It’s more of just having faith, perhaps just like you have faith that the scientific method will provide you with the truth, though I am not sure that is the same faith.

    That’s really my only “beef” with atheists. I don’t care that you don’t have faith, but when you say that you don’t believe in “God” I’d also like for you to explain to me what you mean by that. In a purely philosophical way, it is easier to have a faith in God, whatever He may be (there are tons of possibilities, than to not have faith in God and have to defend against all those interpretations. Again, that is not to say your lack of faith isn’t valid when it comes to the prominent religions’ understandings of God, but to say you don’t believe in God at all seems, well, almost capricious.

    Let me know what you think.

  13. roscoe says:

    Oh, and as for the original post and to all those who say the complexity of life is the proof of some designer, I’d like to throw in a little guy that’s close to home (He was my professor of analysis freshman year of college): Conway’s game of life.

    If you don’t know about it, there are something like three really basic rules (wikipedia it), but from the interaction of those rules, super complex patterns can emerge. It’s just ignorant to say that complexity cannot arise from simple laws of nature…

  14. roscoe says:

    Ok, before I get mauled, I’ll admit to not having read The God Delusion and I’m a tad embarrassed about my comment now. I just read the first chapter-ish from his website and he does a damn good job of explaining everything I just did. I have to go at the moment, but still, I think my comments about faith still stand. I’ll elaborate if you’d like. Thanks.

  15. Atman is Brahman says:

    What you all fail to realize is that reality is infinite. That being the case, there is plenty of time and space for EVERYTHING to come into and go from existence.

  16. you just got pwned says:

    Dude, ok I get the whole probability thing, and I get your point that the small amount of probability doesn’t really matter. As long as their is some sort of probability the chance could happen. Except your missing one thing, the probability of spontaneous life is 0.0. Meaning that it can’t EVER happen. How in the world can you take space, and turn it into a living environment. I believe if life was spontaneous, that it would have come from not even a universe with stars and planets, but space. Space being nothing.
    Whatever tho, so ok life probability is 1^200 or whatever you said. You also have to think of the probability of a god existing. With all the stuff that earth and life is having a god has to be at least 1^25 or something. You obviously know that a god is much more probable then spontaneous life. Anyway, as humans we have only one decision to make, and we have to assume that there is a god based on the massive differences in probability.

    • mywifesnameishonda says:

      Have you ever done any studying or research, ever? Do you have any form of degree from anywhere? I’m a below average college student from a below average state school, but reading your post was like trying to listen to my 4-year-old neice explain how God banned slavery and take her seriously. You sound foolish and seem to be making things up as you go, not to mention your depressing English skills.

  17. edge100 says:

    What is the evidence that the probability of spontaneous life is zero? You just asserted that, by fiat; you’re begging the question.

    You’re just making this stuff up as you go.

  18. Kindoalkun says:

    Using science to “disprove” God is just as distasteful as using it to “prove” God. Until EITHER side has PROOF just STFU please.

    Peace out.

  19. edge100 says:

    Corrected –

    Using science to “disprove” Leprechauns is just as distasteful as using it to “prove” Leprechauns. Until EITHER side has PROOF just STFU please.

    Yeah, sounds just as silly that way.

  20. [...] [3] Flipping Our Coin March 30th, 2007 Article [...]

  21. RobG says:

    Michael, loved the post on coin-toss probabilities, which is why I came here. However, your conclusion doesn’t match the criticism that theists make against Abiogenesis and the spontaneous creation of life.

    If you take the 22 proteinogenic amino acids and combine them in strands of say 150 to well over 1,000 to make various proteins necessary for various parts of a cell. What are the odds that a cell, capable of self-sustainability and self-replication, can be spontaneously combined in the right amount, order, and length for each of the cell’s components to create life? The probability is so high that it is impossible.

    Second you stated, “100% likelihood of some outcome”. That is true, but with life the likelihood of the outcome is being 100% identical. There is only a very small chance when a cell copies itself that it mutates. And when it does mutate, it’s is generally a degradation of the genetic code. Point mutations do not add beneficial genetic material to the genome, they only change existing code. I.e. a bacteria that mutates to become strain-resistant is still only a bacterium.

    These are the true issues that theists have here.

  22. AEKC says:

    I take issue with the authors explanation for why any combination of heads/tails is just as noteworthy as 100 tails (or heads) flipped in a row (even though this is besides his point). True, any given predetermined combination of 100 flips has the same probability of 100 tails in a row. Though 100 tails in a row is also 100% of a single possible outcome. 50 heads and 50 tails is far more likely. This is why one is theoretically justified in being “absolutely astonished” and having reason to believe the coin is “fixed” if 100 tails are flipped in a row. It is a law of probability that as the sample set approaches infinity the ratio of heads to tails will approach 0.5 in the case of a true coin. We are shocked if a series of flips results in an improbable ratio of heads to tails, not the individual sequence of outcomes.

    Also his analogy involving betting on the unique combination of 24 flips is not accurate unless he is suggesting that all betters are drawing from a hat that has one of each unique combination on a piece of paper. If it is similar to the lottery where every one is making a guess, then it is not certain that some one would win. and only an idiot would guess 24 tails. If you wanted to increase your chances of winning you would guess some combination that broke out to a near 50/50 split between the total heads and tails.

  23. Tom says:

    An atheist does believe in God, however that god is not the creator of the universe. It is in the place of that God. Anti-Christ is not against, but in the place of. We are all gods, but there is one God. Emanual = God with us. God gave all authority to His Son until everything comes under His Son’s feet, then Christ will give it back (present it) to God. Why do most people don’t know God, if He is suppose to be ‘with us’. And His Spirit (the Holy Spirit/Ghost) has been given to us as a counselor. Shouldn’t we know God’s presence then? Is His handwriting in creation (nature) anywhere? What is the probability of a dice coming up with a 6 when the side with the 6 is made of lead and the side with the 1 is made of light plastic? What if there are things that are heavily weighted that could not have come about randomly? Look up Bomb Bomb bug of Africa. Very complex chemical reactions and inhibitors. If mixed wrong, then boom… Nothing left to reproduce. Have you ever searched for God sincerely? His Spirit is what convicts our conscience (our wakefulness part of us) either for us or against us. We are able to sear our conscience and not listen. Is this what has happened to Emanual? Do we rely too much on ourselves? Do we really need Him or even want Him?

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