I’m a fan of Michael Shermer, a prominent atheist and skeptic who writes a column on skepticism for Scientific American. But in an article entitled “How Randomness Rules Our World and Why We Cannot See It” he has fallen for a mistake common among atheists today. The mistake involves misunderstanding the relationship between probabilities and the physical world—specifically assuming that the physical world is probabilistic in nature. This is wrong. In fact there is no relationship between probabilities and reality at all. Probabilities relate solely to our knowledge or lack of knowledge of something, and as such can tell us nothing at all about the nature of reality.
On its surface, this may not appear to have much to do with atheology. But if you stick with me you will see that the surface is misleading. Underneath the surface this is all about the nature of reality and therefore it is about what sort of natural worldview, if any, fits with the facts of our existence.
Most atheists today, I would guess, assume a version of naturalism based on scientific realism. The underlying assumption of scientific realism is that correct scientific knowledge is possible and when obtained that knowledge uncovers the fundamental nature of the physical world. I for one think that scientific realism is off base. I don’t think it’s compatible with an evolutionary explanation of the origin of the mind. Since as I have argued elsewhere, naturalism isthe proposition that mind was not present from the beginning but came into existence later, it follows that naturalism requires an evolutionary explanation for the mind’s advent. One purpose of this blog is to try to make the case that scientific realism should be rejected by atheists and advocates of naturalism.
My version of naturalism is based on neurological constructivism, the view that knowledge is a model of the world constructed by the brain simply because it’s useful for survival. As such, knowledge is about usefulness, not truth. Our minds evolved to develop knowledge models of the world based on the application of pragmatic empiricism. If I were to give a one sentence explanation of pragmatic empiricism, I would say that it is the idea that there is no way to verify our knowledge of the world against the world itself, other than to observe its usefulness.
I will write more about neurological constructivism and pragmatic empiricism in the future. I mention them here only to give the reader a bit of context for what follows. If we adhere to scientific realism, we assume that probabilities are inherent in things, and we may even conclude that randomness is inherent to reality.
To help make his case that randomness rules our world, Shermer foolishly introduces the Monty Hall problem.
Imagine that you are a contestant on the classic television game show Let’s Make a Deal. Behind one of three doors is a brand-new automobile. Behind the other two are goats. You choose door number one. Host Monty Hall, who knows what is behind all three doors, shows you that a goat is behind number two, then inquires: Would you like to keep the door you chose or switch? Our folk numeracy—our natural tendency to think anecdotally and to focus on small-number runs—tells us that it is 50–50, so it doesn’t matter, right?
Wrong. You had a one in three chance to start, but now that Monty has shown you one of the losing doors, you have a two-thirds chance of winning by switching. Here is why. There are three possible three-doors configurations: (1) good, bad, bad; (2) bad, good, bad; (3) bad, bad, good. In (1) you lose by switching, but in (2) and (3) you can win by switching. If your folk numeracy is still overriding your rational brain, let’s say that there are 10 doors: you choose door number one, and Monty shows you door numbers two through nine, all goats. Now do you switch? Of course, because your chances of winning increase from one in 10 to nine in 10. This type of counterintuitive problem drives people to innumeracy, including mathematicians and statisticians, who famously upbraided Marilyn vos Savant when she first presented this puzzle in her Parade magazine column in 1990. —Michael Shermer, “How Randomness Rules Our World and Why We Cannot See It”
Shermer’s explanation of why the contestant should switch is incomplete and inadequate. Surprisingly, Marilyn vos Savant and before her Martin Gardiner, who presented an earlier version involving cards in his long-running column on mathematical games in Scientific American, have misled Shermer. His conclusion about the probability calculations which apply in this situation only holds if one stated and two unstated assumptions are true. To arrive at Shermer/vos Savant calculation of probabilities, the contestant must (1) assume that Monty knows the winning door, (2) assume that Monty intended to reveal a losing door no matter which door the contestant initially choose, and (3) assume Monty has no underlying motive when providing the offer to switch doors.
The problem is this: our experts seem to be unaware that they expect the contestant to blindly accept these three underlying assumptions as automatic givens.
True enough, in the problem description Shermer explicitly states “…Monty Hall, who knows what is behind all three doors, shows you that a goat is behind number two….” But just because those of us reading Shermer’s article know this “factual” stipulation, it doesn’t follow that the contestant knows it. Instead, it’s something the contestant is supposed to assume but cannot know with factual certainty.
Why does this matter? It matters because if Monty doesn’t know which is the winning door—or has forgotten—when he reveals door 2 to have a goat, it makes a material difference to the odds which one calculates. In this new scenario, where Monty doesn’t know or doesn’t remember the winning door, the contestant’s “folk numeracy” (as Shermer calls it) would be correct: the odds between the two unrevealed doors are now 50–50.
How can it be that what Monty knows (or doesn’t know) when he reveals a door affects the odds?
Let’s look at it from the contestant’s point of view. Once they pick door 1, two unchosen doors remain: they are either both goats (if door 1 is the winning door) or else one of them is the winner while the other has a goat. If follows that if one of the unchosen doors is now randomly revealed to show a goat, the contestant has learned something new, and therefore ought to recalculate their odds with the remaining doors. Two remain, so the new odds are 50–50 between them. The key word above is randomly.
But if Monty knows which door is the winning door, and if he doesn’t want to reveal that door yet, then if follows that when he reveals door 2 to show a goat, he’s not doing so randomly. He will always have a goat to show, assuming he’s aware of what’s behind each door. In short, Monty is pretending to give the contestant new information but, in fact, he is not.
If the contestant proceeds to miscalculate the odd, it’s not because of faulty “folk numeracy” but rather because they failed to realize that Monty didn’t randomly reveal a door. They have been tricked.
Now, it’s a pretty safe assumption that Monty knows what’s behind each door—but one that is not 100% certain. It can’t be 100% certain. Monty might have forgotten, or got confused, or just misheard when he was told ahead of time. Mess-ups can happen. And perhaps if the contestant noticed uncertainty or anxiety in Monty’s face as he opened door 2, they might suspect Monty wasn’t sure about what was behind the door after all. And this would be pertinent information—if taken as reliable—which should change the odds they now calculate.
Shermer has misunderstood and mischaracterized the Monty Hall Doors problem because he mistakenly believes that probabilities are inherent to physical situations. He fails to realize that probabilities are not “discovered” in the physical world around us, but are the result of judgements we make about what we do or do not know—and even about what we believe others do or do not know.
The Second Contestant
Consider this virtually identical situation, in which there are still exactly 3 possible door configurations: (1) good, bad, bad; (2) bad, good, bad; (3) bad, bad, good. Monty lets the contestant pick a door, and they pick door #1, exactly as in the original. But then, during a commercial break and unknown to that contestant, Monty introduces another contestant—not onstage where the first is, but online—and lets her pick a door. She picks door #3. Monty now reveals to all that door #2 has a goat behind it, just as in the original situation.
Now what should they do if Monty offers each contestant (still unaware of each other or their picks) the opportunity to switch doors?
Well, according to Shermer, their “folk numeracy” would mislead them into thinking it doesn’t make a difference—that the chances are the same with door 1 or door 3. But that would be wrong, according to “expert numeracy”. After all, to quote Shermer again,
Our folk numeracy—our natural tendency to think anecdotally and to focus on small-number runs—tells us that it is 50–50, so it doesn’t matter, right?
According to Shermer, you should in fact jump at the opportunity to switch your choice from door 1 to door 3 because, since door 2 has been revealed as a loser, door 3 has twice the probability of being the winner than does your original choice.
But wait—this analysis holds not just for the first contestant but for the other contestant as well. According to “expert numeracy” both should jump at the chance to switch. The door on the other side really is greener—each has a two-thirds chance of winning if they swap choices.
Obviously, there is something wrong with this sort of “expert numeracy”. Nothing materially has changed about what’s behind the doors in this two-contestant scenario. In both cases, Monty knows which door is the winning door (or more pertinently, the contestants assume he does). In both cases, Monty reveals the middle door to harbor a goat. In both cases, there were originally “three possible three-doors configurations: (1) good, bad, bad; (2) bad, good, bad; (3) bad, bad, good”, but the conclusion that there is an advantage in switching is now false.
What’s different this time is that Monty had only one unchosen door he could reveal. In effect, door 2 was revealed randomly, since the contestants randomly selected the other two doors, leaving it as the only remaining unchosen door. Even if Monty knows what’s behind each door, it doesn’t matter. He’s no longer got a choice.
To viewers watching this on tv, seeing the two contestants, it’s going to be clear that the odds are now 50–50 between door 1 and door 3. But to each contestant, unaware of the, it will have appeared that Monty had two unchosen doors to decide between. Thus if they were savvy, concluding that Monty didn’t just randomly open door 2, that instead he strategically selected it as a remaining goat door, an offer to switch doors would seem to be to their benefit. If one of the contestants was Michael Shermer, he’d be convinced he was doubling his chance of winning.
In fact, the odds that ought to be calculated differ depending on how much you know. If you are one of the contestants, unaware of the second contestant, you make certain assumptions about what you know and calculate the odds based on what you know. But if you are viewing it on tv, thus aware of both contestants, you calculate the odds differently because you know more. And if you are Monty (remembering what’s behind each door) you don’t even have to calculate odd, since you know which door is the winner. (On the other hand, if you are Monty and you’ve forgotten, then you too must calculate odds based on what you believe you know.)
In short, the experts have missed something. They have failed to realize that probabilities differ based on what each individual doing a probability calculation knows.
The Professor
But even this is an insufficient analysis. Imagine that a Professor somewhere has carefully studied Let’s Make a Deal,and in that study has observed that 72% of the time when Monty makes a switch offer, it is to a contestant who has already chosen the winning door.* So let’s go back to our first scenario with the single contestant—but with one difference: our contestant happens to have read about the Professor’s observation.
Does this change the odds for the contestant? You bet it does! (But only if the contestant considers the Professor’s study reliable.)
I hope my point is clear: probabilities only pertain to our knowledge (or lack of knowledge) of the physical world—probabilities do not pertain to the physical world itself, never have and never can. If one wants the simplest possible proof of this, it’s found in the fact that probabilities can differ for each observer.
Consider Let’s Make a Deal again. Up on that stage, Monty knows which door is the winning door—so let’s ask a question: what is the probability that the door he knows to be the winning door is in fact the winning door? For the contestant faced with picking a door and knowing nothing beyond the fact that there are 3 doors, each door must be assigned a 1/3 chance. But for Monty, two doors have virtually no chance and one door is a virtual lock. (It’s not 100%, however, because Monty may have remembered incorrectly, or been misinformed by the show’s producer, or a rare snafu may have resulted in the prize being put behind the wrong door). Monty and the contestant have different sets of knowledge, and so the probabilities differ depending on whose perspective we choose.
Different observers have different knowledge and therefore properly assign different probabilities. And this means simply that probabilities are not inherent in things. “Randomness” does not rule our world, any more than “certainty” rules it.
*This is not implausible. According to Wikipedia, when the New York Times interviewed Monty Hall and discussed “the Monty Hall problem” with him, “Hall clarified that as a game show host he … did not always have to allow a person the opportunity to switch (e.g., he might open their door immediately if it was a losing door, might offer them money to not switch from a losing door to a winning door, or might allow them the opportunity to switch only if they had a winning door).” It depended on his mood. https://en.wikipedia.org/wiki/Monty_Hall_problem, captured Aug 2, 2025
See also: https://dwightlyman.com/articles/aces-to-god/
Post was updated August 2, 2025.
