---

On the Legitimacy of Inquiring into Explanations for Improbable Phenomena
Have we any right to be surprised by any event?
-by Paul Smith, 8/16/99, last rev. 03/13/00

---

"It is fallacious to assert that since some set of conditions must obtain in the universe, the actual set is in no way improbable or in need of explanation." - William Lane Craig [1]

The assertion often gets made in atheist vs. theist dialogues that a student of the dialogue, when evaluating theistic arguments from vast improbability, should evaluate them as if the theistic arguer either intends to deceive or perhaps is ignorant of how mathematics work with regard to probability.  The stronger a theistic conclusion is, the stronger will the suspicion be of ignorance or malice on the part of the theist.  This article does not intend to establish vast improbability for any particular cosmological, geological or biological reality; rather, it intends merely to show that two popular arguments often employed against this sort of theistic argument are either trivial or fallacious.

The two arguments that will be examined are:

1. The fact that an event has occurred is, in and of itself, justification for declaring the probability of its having occurred irrelevant.
   
   ...and...
   
   

2.  Vastly improbable events happen all the time, therefore, the theist has no right to incredulity over a vastly improbable event that happened to give rise to our existence; it is one among many.

ŻŻŻŻŻ

Let's start with the first argument - "The fact that an event has occurred is, in and of itself, justification for declaring the probability of its having occurred irrelevant ."  Some critics object to the use of statistical/probabilistic arguments on the grounds that they may not be used to discern likely causes of an event after the fact. The reasoning goes something like this:  "X has obtained [happened], therefore the probability that X did obtain is 1:1".  This argument bears a close resemblance (but is not identical) to some of the various Anthropic Principles; it basically asserts that since we are here, it is meaningless to inquire how likely it is that we would be here.  However, this response misses the fact that what is being asked is not what the odds are that X did happen, but rather what the odds are that X would have happened, given our knowledge of initial conditions and laws that might govern the outcome. (In many cases, we have no reason to believe that the initial conditions or governing laws must have taken any specific form or value at all - that is, we know no reason why they should be what they are.)  

Perhaps an anecdotal example would be useful at this point.  Let's say our "X" is the phenomena of my brother winning the state lottery for 52 consecutive weeks.  Given the way my state runs its lottery, I would be hard pressed to think of a way that the system could be cheated; those of you who participate in such "capital ventures" are probably familiar with the routine I'm going to describe.  Tickets are purchased before the drawing, at which time an air-blown hopper with numbered ping-pong balls is turned on.  At this point that week's volunteer (usually a rather un-telegenic associate of some accounting firm) steps up to the device with all the balls jumping wildly around in it, and begins to sequentially open each of the six tubes protruding from the top of this single-function contrivance.  When this process is completed, six of the numbered (one to forty-something) balls will have emerged triumphant.

So now let us imagine that my brother has in fact managed to pick these winning numbers 52 weeks in a row, and let us call this phenomenon "X."  My question to the reader is this - would you in all honesty argue that since X has in fact occurred, it is therefore meaningless to inquire as to the likelihood of X having happened?   Mind you, this is not a situation where intelligent intervention is the most obvious explanation, or even easy to postulate (although I'm sure I'll eventually see some far-out ideas on how it could be done... don't try this at home, kids!)  Here it is obvious by analogy that the past occurrence of phenomenon X is no justification for declaring probabilities irrelevant.  I would seriously question the intuition of the person who would give my brother the benefit of the doubt here, too!

Of course it is true that our existence is a necessary precondition of our ability to express amazement at our existence.  There's a certain "duh" factor in this realization; Craig illustrates this principle by pointing out that if a person before a firing squad happens to get missed by all the shooters, he shouldn't be surprised that he isn't observing that he is dead, but he has every right to be surprised that he is observing that he is alive, and he has every right to seek an explanation for this observation!  Now if somebody finds the fact of the universe's existence and of its conformity to the myriad narrow requirements for the existence of intelligent life to be sufficient justification for prohibiting questions of why and how and how likely this all is, so be it.  However, in all honesty I have a hard time imagining that people really find this sort of reasoning to be intellectually satisfying; it does not seem that people employ this sort of decision making methodology in their daily lives.  The fact is that all events beg explanations; it just happens that some are explained quite easily by regularly observed processes (or unintelligent causes).  In short, the anti-theist argument discussed here is unimpressive because the happening of an event is a prerequisite to us inquiring why it happened.  The critic is for all practical purposes trying to use the event as its own explanation.  This may provide a desperate person with a temporary escape from theistic conclusions, but we don't do this in daily life, we don't do this in the academic disciplines, and we shouldn't be impressed because someone wants do it with the big questions of life.

ŻŻŻŻŻ

Now let's take a look at the second objection - "Vastly improbable events happen all the time, therefore, the theist has no right to incredulity over a vastly improbable event that happened to give rise to our existence; it is one among many."  Well, have we any right to claim surprise over any particular event occurring?  I hope the answer to the above will become apparent in the following process.  William Dembski uses an example of a series of 100 coin flips in making a point about the role of specificity in making judgment calls about chance and design inferences [see "further reading"].  To this example in general, Steven Jay Gould remarks "give me a million years and I'll flip a hundred heads in a row more than once [2]."  What is interesting to me about this statement is not Dr. Gould's opinion about the likelihood of doing such a thing; rather, it is the nature of his answer.  It strikes me as obvious that in suggesting an intuitive plausibility in attaining success against these odds given enough time (say, a million years), Gould implicitly admits that probabilities are anything but irrelevant to explanations of origins.

Of course, the argument from small probability recognizes that it is impossible to flip a coin 100 times without causing some astronomically unlikely event to happen.  (Try it yourself; flip a coin 100 times, record the result, and calculate the probability of having gotten that result.  Voila! - you've just caused an astronomically unlikely event to happen.)  I do suspect that design proponents are more aware of this than their critics would like to think, and I hope that the reader of this article will walk away with a new appreciation for the difference between any event and a specified event. To illustrate this a bit more vividly, an experiment will ensue.

The experiment is this: We'll use a computer to simulate the coin-flipping over time (our simulation so far consists of 2 runs -- 100 million and 3 billion attempts, respectively -- at a series of 100 coin flips.)  The "goal" is, per Gould's boast, to "flip" 100 straight heads in the space of a million "years." Let's call this our specified result.  Assuming Dr. Gould could maintain a rate of two flips per minute for the duration of his million-year period (we'll grant him the ability to flip during meals, bathroom breaks and sleep time for this example), this would allow him 15,778,493,280,000 coin flips during his marathon.  The experiment has not yet exhausted this search space, although with distributed computing it is entirely possible that this could be done in the near future [3].  In the meanwhile, though, it's important to remember that the claim being countered here - "Vastly improbable events happen all the time..." (you remember the rest, right?)  Now mathematically, our series of 100 heads has the exact same probability as any other series of flips; indeed, even the first series of flips that actually did occur was every bit as unlikely. (This admission should please our critics.)  With so many attempts, we should downright expect to hit our "chosen" result, n'est- ce- pas?  I mean, it does have the same probability of any other result.

So why should specificity matter? Good question. Let's look at a real-world example, and put the question back in the lap of the misguided critic. Given the usage of some opponents' comments that "probability does not have a memory of past events," it would logically follow that a particular series that begins with 50 heads is no less likely than one that begins with only one head. This is not the same as asking for the odds of a particular series of only 50 flips, according to the critics' logic; were I to repeat the series of 100 flips 10 million times, there is no mathematical law making my desired series (which happens to start with 50 heads) any less likely than one starting with, say, "THTHHHTHHTHTH....".

Now I must confess that that preceding statement is mathematically true as it is phrased, but the way the statement is used by some opponents of the design argument demonstrates a profound misunderstanding of the role of succession and specificity as it is used by design proponents.  In other words, I'm blatantly cheating here to illustrate where our critic takes a wrong turn.  In that spirit, here's a phenomenon that proponents of "vastly improbable events happen all the time"-type logic should be able to explain to us: In the following table (see below), according to this particular line of logic, I would expect a series starting with 28 Heads to be equally probable to series starting with 13 Heads. Therefore, the distribution of actual successes in the various "number of matching coins before failure" rows in the table ought to be fairly uniform, right?  But this is obviously not the case. Why not?

Should some contentious type respond with the only viable argument (and the correct one) against my tongue-in-cheek example, he will have in the process affirmed our thesis and therefore capitulated on his original objection: in short, a specific highly-improbable event is indeed quite different from just any highly-improbable event, at least as far as the results below are concerned. Incidentally, as of this writing I have run over 7,250,000,000 series, (3,100,000,000 combined are recorded here), but have never had my pre-specified series of 100 heads obtain despite having had 7,250,000,000 other, equally unlikely results actually occur. Can the critic explain why this is? After all, there is no mathematical law prohibiting a match on the very first attempt.

I would enjoin people reading this article to keep in mind that hitching this reasoning onto any particular observable phenomenon requires some homework first.  Almost every school of political thought will have articles within their fundraising and polemical literature that include phrases like "Lies, Damn Lies, and [our opponents'] Statistics."  I have encountered my share of bad statistics in theistic and anti-theistic arguments alike, and would hate to see some reader of this article become another statistic himself!  This article is not intended to "drive home" the notion that any particular phenomenon we observe is necessarily unexpected.  Rather, I simply hope that I have illustrated that there is validity in expressing surprise that certain phenomena have obtained, and that the request for an explanation for these phenomena is a legitimate request for one to make. 

-Paul C. Smith

NUMBER OF MATCHES AT START	ODDS AGAINST IN-SEQUENCE MATCH	ACTUAL OCCURENCES IN 100,000,000 TRIES	ACTUAL OCCURENCES IN 3,000,000,000 TRIES
1	2:1	49,998,644	1,499,981,898
2	4:1	25,008,000	750,226,833
3	8:1	12,498,202	374,942,353
4	16:1	6,253,079	187,585,744
5	32:1	3,120,808	93,628,385
6	64:1	1,561,868	46,848,474
7	128:1	777,875	23,338,021
8	256:1	390,271	11,709,326
9	512:1	196,486	5,895,102
10	1,024:1	97,770	2,933,931
11	2,048:1	48,267	1,447,826
12	4,096:1	24,650	739,793
13	8,192:1	11,572	347,217
14	16,384:1	6,593	197,739
15	32,768:1	2,718	81,519
16	65,536:1	1,670	50,067
17	131,072:1	666	20,027
18	262,144:1	491	14,663
19	524,288:1	190	5,719
20	1,048,576:1	48	1,431
21	2,097,152:1	84	2,502
22	4,194,304:1	36	1,072
23	8,388,608:1	-	-
24	16,777,216:1	12	358
25	33,554,432:1	-	-
26	67,108,864:1	-	-
27	134,217,728:1	-	-
28	268,435,456:1	-	-
29	536,870,912:1	-	-
30	1,073,741,824:1	-	-
31	2,147,483,648:1	-	-
32	4,294,967,296:1	-	-
33	8,589,934,592:1	-	-
34	17,179,869,184:1	-	-
35	34,359,738,368:1	-	-
36	68,719,476,736:1	-	-
37	137,438,953,472:1	-	-
38	274,877,906,944:1	-	-
39	549,755,813,888:1	-	-
40	1,099,511,627,776:1	-	-
41	2,199,023,255,552:1	-	-
42	4,398,046,511,104:1	-	-
43	8,796,093,022,208:1	-	-
44	17,592,186,044,416:1	-	-
45	35,184,372,088,832:1	-	-
46	70,368,744,177,664:1	-	-
47	140,737,488,355,328:1	-	-
48	281,474,976,710,656:1	-	-
49	562,949,953,421,312:1	-	-
50	1,125,899,906,842,620:1	-	-
51	2,251,799,813,685,250:1	-	-
52	4,503,599,627,370,500:1	-	-
53	9,007,199,254,740,990:1	-	-
54	18,014,398,509,482,000:1	-	-
55	36,028,797,018,964,000:1	-	-
56	72,057,594,037,927,900:1	-	-
57	144,115,188,075,856,000:1	-	-
58	288,230,376,151,712,000:1	-	-
59	576,460,752,303,423,000:1	-	-
60	1,152,921,504,606,850,000:1	-	-
61	2,305,843,009,213,690,000:1	-	-
62	4,611,686,018,427,390,000:1	-	-
63	9,223,372,036,854,780,000:1	-	-
64	18,446,744,073,709,600,000:1	-	-
65	36,893,488,147,419,100,000:1	-	-
66	73,786,976,294,838,200,000:1	-	-
67	147,573,952,589,676,000,000:1	-	-
68	295,147,905,179,353,000,000:1	-	-
69	590,295,810,358,706,000,000:1	-	-
70	1,180,591,620,717,410,000,000:1	-	-
71	2,361,183,241,434,820,000,000:1	-	-
72	4,722,366,482,869,650,000,000:1	-	-
73	9,444,732,965,739,290,000,000:1	-	-
74	18,889,465,931,478,600,000,000:1	-	-
75	37,778,931,862,957,200,000,000:1	-	-
76	75,557,863,725,914,300,000,000:1	-	-
77	151,115,727,451,829,000,000,000:1	-	-
78	302,231,454,903,657,000,000,000:1	-	-
79	604,462,909,807,315,000,000,000:1	-	-
80	1,208,925,819,614,630,000,000,000:1	-	-
81	2,417,851,639,229,260,000,000,000:1	-	-
82	4,835,703,278,458,520,000,000,000:1	-	-
83	9,671,406,556,917,030,000,000,000:1	-	-
84	19,342,813,113,834,100,000,000,000:1	-	-
85	38,685,626,227,668,100,000,000,000:1	-	-
86	77,371,252,455,336,300,000,000,000:1	-	-
87	154,742,504,910,673,000,000,000,000:1	-	-
88	309,485,009,821,345,000,000,000,000:1	-	-
89	618,970,019,642,690,000,000,000,000:1	-	-
90	1,237,940,039,285,380,000,000,000,000:1	-	-
91	2,475,880,078,570,760,000,000,000,000:1	-	-
92	4,951,760,157,141,520,000,000,000,000:1	-	-
93	9,903,520,314,283,040,000,000,000,000:1	-	-
94	19,807,040,628,566,100,000,000,000,000:1	-	-
95	39,614,081,257,132,200,000,000,000,000:1	-	-
96	79,228,162,514,264,300,000,000,000,000:1	-	-
97	158,456,325,028,529,000,000,000,000,000:1	-	-
98	316,912,650,057,057,000,000,000,000,000:1	-	-
99	633,825,300,114,115,000,000,000,000,000:1	-	-
100	1,267,650,600,228,230,000,000,000,000,000:1	-	-

---

FOR FURTHER READING:
See William A. Dembski's "Randomness by Design" and "The Explanatory Filter..." for treatments of the mathematical aspects of these issues that are beyond the purview of this article.  The curious may also peruse more of Dembski's papers, as well as other authors' treatments of this subject matter from a theist perspective, at http://www.origins.org/menus/design.html.

FOOTNOTES:
[1] William Lane Craig in "THE TELEOLOGICAL ARGUMENT AND THE ANTHROPIC PRINCIPLE," emphasis added.  See that article for a far more systematic and rigorous refutation of the arguments against Design from the Weak Anthropic Principle.

[2] Steven Jay Gould, The Panda's Thumb, p.218

[3] If anyone would care to assist in this project, all that is required is a computer running Windows 95, 98, 2000 or NT, and Microsoft Excel 97 or higher.  A Pentium-450 computer should be able to try about 164,000 series per second; a few hours of otherwise idle processor time could help significantly! Send E-mail to webmaster@theism.net to participate.  I will also be happy to provide the most recent VBA source code to anyone wishing to examine it.