WMD

    Words of Mass Dissemination

Last night, I drew cards from a deck. The first card was the Ad. Then came 2d. Then 3d, 4d, 5d, 6d, 7d, 8d, 9d, 10d, Jd, Qd, Kd. The next card was the As, followed by 2s, 3s, 4s, 5s, 6s, 7s, 8s, 9s, 10s, Js, Qs, Ks. And then, believe it or not, the cards ran through hearts and clubs, in order again. The probability of that happening is unimaginably low. In particular, the probability of getting that Ad on the first card was 1/52. The probability of getting the 2d next was then 1/51. So the probability of getting just the first two cards (Ad followed by 2d) was 1/52 x 1/51 = 1/2652. Following through this reasoning, the probability of drawing the entire deck in the order that I just described is about 1.24 x 10^-68. Wow. That’s low. If you shuffle and draw once every second (quite a feat!), you’ll be reasonably likely to draw the cards in this order after about 10⁶⁰ years. (The current age of the universe is believed to be around 10⁹ years.)

Well, with that information, you’re probably a bit skeptical of my story. But trust me, it’s true. What do you make of it? Almost certainly, somebody put that deck in order before I picked it up.

But must we really draw that conclusion? Suppose, instead of the order that I reported above, I told you that I drew the cards in this order: 4c,8d,10h,2h,8s,Ad, and so on. What is the probability of that result? Well, it is exactly the same. Every possible ordering of the cards has the same extremely low probability as every other. (The probability of getting the 4c on the first draw was 1/52. The probability of then getting 8d was 1/51, etc….)

What is the real difference between the ordered and unordered decks? Why is it, after all, correct to say that the first ordering is somehow unlikely, while the second is unsurprising. Many will say that the relevant distinction here is the distinction between a random and non-random sequence of cards. So the correct question to ask is not ‘what is the probability that we draw that particular sequence?’ but ‘what is the probability that we draw a non-random sequence?’ The first sequence (the ordered one) is non-random and the second is random. The random sequences are far more likely than the non-random ones. Hence we are correct to believe that the first result, but not the second, is in some need of further explanation.

There are two features of this argument that are in need of further examination. The first concerns the manner of defining the notion of a non-random sequence. The second concerns the inference from ‘low probability’ to ‘in need of explanation’.

What makes the sequence Ad, 2d, 3d, 4d, 5d, 6d, … non-random and the sequence 4c, 8d, 10h, 2h, 8s, Ad random? There are different approaches to this question, but the philosophical disputes about the nature of randomness need not detain us, for they all rely on some way of identifying the relationships amongst the cards, so that judgments can be made about the appearance of ‘non-random patterns’. For example, the sequence Ad, Ah, As, Ac, 2c, 2h, 2d, 2s, 3s, 3c, 3d, 3h, … is non-random. Why? Because any card with an ‘A’ on it is taken to be ‘the same rank’ as any other such card, and so on for ‘2′, ‘3′, etc…. Hence there is a ‘pattern’ in this sequence, and it is, essentially, the appearance of ‘patterns’ that makes a sequence non-random.

If, on the other hand, the card with an ‘Ad’ on it and the card with an ‘As’ on it were not taken to be related to one another in any particular way — and so on for the others — then the sequence above would not be considered to have any ‘patterns’. In other words, the cards themselves ‘know nothing’ about patterns. They are just as likely to produce one sequence as another. But given the meaning that we attached to the symbols on the cards, some of these equally likely sequences are random, others not.

The general idea, then, is that the set of all non-random sequences — given some account of what counts as random, and in particular what counts as a ‘pattern’ — is much smaller than the set of random sequences. Hence, if we get a non-random sequence, we can assert, with some truth, that something with low probability occurred, something (namely, a non-random sequence) that was ‘unexpected’, whereas if a random sequence occurs, then we can say that something with high probability occurred, something (namely, a random sequence) that was ‘expected’. (Of course, every particular sequence has the same probability as every other.)

The point here is quite analogous to the following situation: imagine a jar with 10,000 black jellybeans and 2 red jelly beans. Your friend is about to reach in and grab a jelly bean for you. It comes out red. Well, on the one hand, each jelly bean in the jar had the same probability of being chosen as every other. So if we describe the result as ‘having grabbed this particular jelly bean’ (as opposed to one of the other jelly beans in the jar), then yes, it has very low probability (1/10,002), but so did every other possible result. But, on the other hand, if we describe the result as ‘having grabbed a red jelly bean’, then we can characterize the ’specialness’ of our result in a way that does not apply to every other possible result: Given the proportion of black to red jelly beans, we would expect to get a black jelly bean, and yet we got a red one.

Does this unexpected result demand some explanation? (I have now reached the second feature of the main argument that needs some examination, namely, whether ‘low probability’ implies ‘in need of explanation’.) We have two options, here. If we say that the result does not require explanation, then we are willing to say ‘it was just luck’. If we can repeat the experiment many times, we can even test that hypothesis, to some extent. (Replace the jelly bean, shake up the jar, and draw again. Repeat many times. If the friend keeps grabbing red out of the jar, then something funny is going on. If the friend grabs vastly more black then red, then our first draw of the red was probably ‘just luck’.)

On the other hand, we might be inclined to search for an explanation. What might prompt this search? That is, what might incline one to say that ‘it was just luck’ is not sufficient? I submit to you that low probability is not enough. Low probability events happen all the time. Flip a coin ten times. The sequence of results that you got (whatever you got!) has probability less than 1/10^th of 1 percent. But nobody thinks that your result is in need of any further explanation. It was just luck. Clearly what makes the difference between any old low probability event and the low probability events that need explanation has something to do with our first point, above, namely, that the low probability event in question can be characterized as ‘unexpected’ relative to some larger class of ‘expected’ results.

Not just any such characterization will do, however. So suppose that you flip that coin ten times. You get: H,H,T,T,H,T,H,T,T,T. I am inclined to say ‘just luck’. But then you point out to me that this sequence is in fact quite special, for it is a member of a small class of sequences with the following property: the longest sequence of tails is half the total number of tails. Very few of the possible sequences have this property. We would have expected to get a sequence without this property, and yet we got one with it. So doesn’t this result need some explanation?

Of course, it does not. What has gone wrong here? Why does the sequence of heads and tails above not need an explanation, and yet the sequence of cards with which I started this discussion need an explanation? Let’s return to the jelly beans to get a clue. Suppose, in addition to the information I already gave you, I mentioned to you that my friend (who will draw the bean) knows that I hate black jelly beans, and for whatever reason wants to make me happy. With that information, we now have some reason to wonder whether the result (a red jelly bean) was not brought about intentionally somehow. In other words, if we have reason to believe that the results in the unexpected set of results are picked out as special by somebody, or something, involved in the production of the result, then there are good reasons to think that if one of these unexpected results occurs, it could very well have been not ‘just luck’.

This principle applies to the two other cases we have considered. Why should we say that the sequence H,H,T,T,H,T,H,T,T,T is ‘just luck’ even though it is a member of the rather small class of sequences whose the longest sequence of tails is half the total number of tails? Because there is no reason to think that the process or people involved in flipping the coin picks out this property as special. Why, on the other hand, should we say that the sequence Ad, 2d, 3d, 4d, 5d, 6d, 7d,… of cards drawn from a deck is not ‘just luck’? Because there are plenty of people in the world, who handle decks of cards, who think that this sequence is special, or more generally who consider the cards to be related to one another in such a way that this sequence is non-random. So the most likely story, in this case, is that one of those people (maybe me!) put the deck in this order before I ever drew a card from it. In other words, there is nothing intrinsically special about this sequence. It is special because some relevant people take it to be so. For suppose that, instead, we happened to ‘order’ decks of cards like this: 4c, 8d, 10h, 2h, 8s, Ad,…. Then drawing the cards in that order would raise the suspicion that somebody produced that result intentionally, while drawing the cards in some other order (e.g., Ad, 2d, 3d, 4d, 5d, 6d, 7d,…) would not arouse the same suspicion.

This whole discussion is, of course, a thinly veiled description of one of the disputes between advocates of ‘intelligent design’ and their opponents. My intention here is to explain why there is an unavoidable stalemate between them, on this point. I am adamantly focused on just this narrow point. There are many other disputes between them. For example, some advocates of intelligent design make empirical claims, or offer interpretations of empirical evidence, that are disputed by some evolutionary biologists. I am no expert on such matters, and set them aside as irrelevant to the present concern, which concerns probability and explanation.

The analogy goes as follows. Advocates of ‘intelligent design’ argue that certain events that have occurred (e.g., the appearance of life on earth) have extraordinarily low probability. For example, they argue that of all of the possible ways of combining atoms, there are only a very few that could ever have resulted in the formation of the sorts of molecule required to support life. Hence the appearance of life on earth was an event of extremely low probability.

But of course, in perfect analogy with the case of drawing the cards from a deck, any particular combination of atoms is a low-probability event — there are many, many, atoms int he world, and they can combine in many, many ways. (The various possible combinations might not all have the same low probability in this case, because there could be physical laws at work that favor some combinations over others, but the general point is the same, unless, of course, those laws make the combinations that have actually occurred overwhelmingly likely. In that case, however, we can just discuss the initial conditions in the universe that gave rise to the result, and a similar discussion would take place. If, finally, the laws are such that any initial conditions would lead to life, then the discussion would probably look very different.) So in order for the particular combination that occurred to be in need of some explanation (other than ‘it was just luck’), it must (i) be a member of an ‘unexpected’ set of results in the sense described above, and (ii) the results in the unexpected set of results must be picked out as special by somebody, or something, involved in the production of the result.

Let’s assume (though the matter is disputed) that the set of possible combinations of atoms that would lead to life on earth is so small (relative to the set of all possible combinations) that the occurrence of life on earth by now has extremely low probability. Hence condition (i) above is satisfied.

(We must say ‘occurrence of life on earth by now‘ because the atoms get more than one chance to combine in the right way. So we have to make some estimate on how many chances they get, say, per year, and how many years they would have had, i.e., how many total chances that they would have had. If the probability that any single event of combination would lead to life is low enough, this total number of chances can be quite high, and yet the probability that any of those chances would have resulted in life might still be quite low. For example, give a monkey a billion chances to type randomly on a keyboard, and the chance that any one of the results is the first page of Hamlet is still unimaginably small, around 1 in 10⁹⁰.)

But what about condition (ii)? Here, alas, we are destined for stalemate. If you antecedently believe that the production of life is picked out as special by something or somebody involved in the production of chemical events, then the occurrence of this special event gives you some reason to wonder whether it was not just ‘by luck’. But if you antecedently believe that the production of these chemical events occurs via processes that in no way pick out life as special, then the correct answer to ‘Why did this low probability event (life) occur?’ is ‘by luck’.

I don’t see any relevant difference between this situation and those described above involving coins and cards. If you do, please send me a comment. If your reaction to what I’ve said here is that ‘the probability of life is so unimaginably small that I just don’t see how it could have happened by chance’ then please reread. If your reaction is that you independently believe that God exists and therefore find it quite likely that God is the origin of human life, you have not (necessarily) disagreed with anything I’ve said. Similarly, if you believe that life is a chance occurrence, that it happened ‘by luck’ (lucky for us!), that there is no point to our existence, etc., then again you have not (necessarily) disagreed with anything I’ve said. If you are fascinated by the prospect of bringing about an event yourself with incredibly small probability, you can be cured. Shuffle 10 decks of cards together, then draw all 520 cards. Your result has probability 1 in:

17610403417821111561460117109879578542911478279626
82515835641902026635013531629049250745289405517992
17492000052911460700851554388506384179657281543248
46728407158333741072487686882930286404967263861328
00905110061351548170024622024983851028980182317268
95681397709425453375071010486886852728967058125963
62498604753026067474836458847095056257533018986923
32700214042981663219274624546482874260801256152782
44807894746803821196970541914927652072207588214838
87707171135570628153647781446524174581916029042277
69668577211157141090782282183940792659802623387773
74065758813337082961068321380721337829277256701491
65229214676654388845288996613236416988679986260143
97661325645202793456690465818078307134319818775325
25575007083897838744272967188911132035890748607857
40418626069545051663666604213934440278998174513866
31912726243989647203455807340483476375101139676265
91761063168892577208286546625327996720369649293722
53320946090105491544711017929034021882046954442464
94414641879321328704507798451597203731365688705702
00042478970118285935361664694673759165844683475607
68565149696000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000