# Is using the law of large numbers to dismiss the significance of improbable event a valid form of inference?

Suppose I decide to play poker today and I'm dealt a royal flush. I then go back to the convenience store and win the lotto. While I'm going to pick up my prize, a speeding car barely misses me. Suppose on the previous night, I was in a depressed mood, and I asked God to pray for me. Today, given that it is my luckiest day ever, I am convinced that I received divine guidance. After all, the probability of this happening without divine guidance is extremely and incredibly low.

Skeptics will look at this event, smile, and say "Well, at some time or another, events like these happen. Rare events happen all the time. It is thus no surprise that given the large numbers of opportunities there are for meaningful events to happen, even the most ridiculously improbable of events can happen."

Although I had always accepted this kind of reasoning, when observing it a bit closely, there seems to be a fallacy here. The fallacy seems to be that this strategy can be used to dismiss any improbable event as being the result of design.

When Alfred Wegener noticed that continents' outlines seemed to match each other and proposed the theory of continental drift, an extremely ardent skeptic could have said "Well, given the trillions of planets that exist in the universe, sooner or later, one of these planets had to have continents that seemed to match each other's outlines by chance. That planet could have been earth. As such, we cannot rule chance out." Assuming that this sounds ridiculous to you, I fail to see how the logic of this argument is any different from the previous one.

Note my question is about whether this specific inference is valid. I am not interested in knowing whether there are other reasons that explain why the continental drift theory makes more sense than God sending down a sign to me. There surely may be. But this specific form of argument is either valid or not. So is it?

• +1 To be honest, I’m surprised how downvoted this question is. There may be confusion the answers address well, but speaking as a mathematician, there is a basic idea underlying this question that is both important and interesting. For example, does the anthropic principle simply end inquiry on many particularities about the origin of the universe? Which ones does it apply to and why not others? Commented Jan 1, 2023 at 9:44
• @thinkingman, consider that one could invert one's sense of all those lucky-day miracles to produce an opposite argument. I.e., you were not unlucky enough to lose the poker game and the lottery, to get hit by a car, etc. so that "shows" that there is not an intentional force bringing harm to you. So the impression you would have, on account of your "good" luck (that there is an intentional force helping you), might be construed as a subjective interpretation of your circumstances. Commented Jan 1, 2023 at 13:03
• No major theistic religion teaches, in the limit, that we know much, if anything, about God's will, and what they do profess is then very general, such as, "God will promote good and suppress evil." Then we have seeming counterexamples of good being defeated, and evil flourishing. "Commonsense" impressions of divine favor in our anecdotal lives thus turn out to be impious on a religious level, or arrogant even (we might think that unsupported goods weren't, it turns out, actually good, but we must be good, after all, if we were given such support!). Commented Jan 1, 2023 at 13:07
• And so finally, turning faith in God into a game of probabilistic arguments is also impious on its face. Whether or not God exists, or whether or not we are justified in believing such a thing, does not depend on word games, but much of popular apologetics on the Internet comes out to such playful depravity (the self-proclaimed believer trying for "gotcha!" moments in debate with so-called unbelievers). "With their lips they do Me honor, but their hearts are far from Me," as they say... Commented Jan 1, 2023 at 13:11
• @JustSomeOldMan Yeah some people just don't seem to understand the question. A pity
– user62907
Commented Jan 1, 2023 at 17:01

You say that the form of argument you refer to is either valid or it isn't- that's a false premise. That form of argument is valid in some circumstances but not in others.

Let's consider the two example you cite in your question. Continental drift can be explained in a way that is consistent with our understanding of how the world works. The idea that divine intervention accounts for your lucky day cannot be. Moreover, there is more evidence to suggest continental drift than just the matching of the shapes of the continents.

So, yes, if the idea of continental drift had been proposed purely based on matching outlines, you might have argued that the effect was just a coincidence. It would then have been down to the proponents of the theory to substantiate their idea with other evidence.

The law of large numbers means that you should bear in mind the possibility that some effect is just a random result- it does not mean that all effects are random results.

It is a mistake to apply guidelines indiscriminately, without considering the context.

• Real life and real science require parsimony and attention to detail, but that's not as easy or popular as soundbites and generalizations. Commented Dec 31, 2022 at 21:44
• @Frank hear hear. Commented Dec 31, 2022 at 22:59

A theist who says, "God probably caused your streak of good luck," and the scientist who says, "The coastlines probably line up due to continental drift," are not using the same concept of probability in their explanations, I suspect. There are topological reasons to expect "counterintuitive"/"unlikely" correlations between coastlines (consider the famous quip that, to a topologist, a doughnut and coffee cup are the same shape), so the local resolution of a discrepancy in probability has a relatively certain background that it appeals to (the relative certainty of topology).

By contrast, the theistic "explanation" for my good-luck streak has no stable mathematical grounding. It doesn't even seem to have a stable religious grounding (Christianity, for example, forbids gambling, does not favor denigrating God as "probably real" rather than necessarily so, and does not provide any reason for us to believe that God will help us for some discernible reason on any given day instead of another); and even more acutely, will fly in the face of counterexamples by the by, revealing itself as anecdotal at best, conniving at worst.

So I imagine that the law of large numbers applies as it does to the theistic explanation of "good luck," as a premise in a counterargument, in a way that it does not apply to the scientific explanation of coastlines; whatever way it does or does not apply to the theory of continental drift will be caught up in the whole ensemble of mathematical considerations (e.g. topology) that go into plate tectonics generally.

EDIT: Consider what happens if we press on beyond the immediate probability claim in either case:

1. The theist says, "It is improbable that you would have had such good luck, had God not favored you today." You then ask, "How did God improve my luck today?" All that the theist can say is, "God used Its ultimate power to rearrange your local contingencies, to give you the desired outcome." An explanation-from-omnipotence is either a trivial explanation or no explanation, as such, at all.

2. The scientist says, "It is improbable that the coastlines would line up were it not for continental drift." You then ask, "How does continental drift explain the harmony of the coastlines?" The scientist can use a wide range of geometrical/topological/dynamical descriptions to explain how the process would work. They can bring up the continuing observed role of magma/lava emissions underwater, in the production of new islands (this is relevant to the thesis that the continents are drifting with tectonic plates that are themselves moving about in part on account of underwater/underground magma cycles). In turn, geothermal processes can be taken into consideration; so too the role of natural fission reactors in geological history, the ways in which various elements interact and redistribute heat/motion; etc. The scientist can, in fact, keep going further and further, deeper and deeper, to justify what is even so a much more general statement (the theory of continental drift) than a statement about the "good luck" I happened to experience on a random day.

Littlewood's Law

Littlewood defines a miracle as an exceptional event of special significance occurring at a frequency of one in a million. He assumes that during the hours in which a human is awake and alert, a human will see or hear one "event" per second, which may be either exceptional or unexceptional. Additionally, Littlewood supposes that a human is alert for about eight hours per day.

As a result, a human will in 35 days have experienced under these suppositions about one million events. Accepting this definition of a miracle, one can expect to observe one miraculous event for every 35 days' time, on average – and therefore, according to this reasoning, seemingly miraculous events are actually commonplace.

A synopsis of Littlewood's thesis:

We experience 1 event every second.

In 1 minute we, ergo, experience 1 × 60 = 60 events (60 seconds in q minute)

In 1 hour we, ergo, experience 60 × 60 = 3600 events (60 minutes in 1 hour)

In the 8 hours per day we're awake we experience 8 × 3600 = 28,800 events.

In how many days can we experience 1 milliom events? 1,000,000 events ÷ 28,800 events/day = 34.7 days (in 35 days, rounding up).

A miracle, defined as an event with a 1 in million chance of occurring, happens once every 35 days. 🙂

You may of course disagree with Littlewood's analysis, specifically whether miracles are rarer/commoner than 1 in a million and our experience rate (how many events do we perceive per second?).

As for stuff like tectonic plates being just a coincidence, you have to realize that scientific claims are a) investigated further to rule out the possibility of such correlations being nothing more than a coincidence (pure chance) and b) it isn't only the correlation (in this case the jigsaw puzzle like fit of the continents) that geologists use to develop an acceptable hypothesis (even the flora and fauna distribution seem to corroborate the tectonic plate hypothesis)

The same argument applies to all similar cases - even the rarest of the rarest can happen by chance and so we, as of routine, have to rule out chance and I've provided a rough sketch of how that's done (vide supra).

Also read up on the Null Hypothesis (statistics).