Are The Dice Loaded?

I am currently reading Universes (1989) by John Leslie, Professor Emeritus of Philosophy at The University of Guelph, Ontario, Canada. The book, praised on the back cover by Antony Flew and Quentin Smith, discusses the issues surrounding the “fine-tuning” of the constants of nature, initial conditions, and even the forms of the laws of nature themselves to permit the existence of observers. I will not go into details of the fine-tuning here – readers are referred to “The Anthropic Cosmological Principle” by Barrow and Tipler.

This is a huge and hugely controversial area and I don’t want to bite off more than I can chew. (Leslie: “The ways in which ‘anthropic’ reasoning can be misunderstood form a long and dreary list”). Instead, I want to consider a single point made by Leslie, in response to the following quote from M. Scriven’s “Primary Philosophy” (1966):

If the world exists at all, it has to have some properties. What happened is just one of the possibilities. If we decide to toss a die ten times, it is guaranteed that a particular one of the possible combinations of ten throws is going to occur. Each is equally likely.

The argument is as follows: we cannot deduce anything interesting from the fine-tuning of the universe because the actual set of constants/initial conditions is just as likely as any other set. It is this claim (and this claim only) that I want to address, because I found Leslie’s treatment to be calling out for an example.

A casino has a game where a die is thrown, and it is advantageous for a player to throw sixes. The casino, naturally, is concerned about cheating players. On top of their usual security measures (cue clip from ‘Ocean’s Thirteen’ – security cameras linked to a computer that can distinguish genuine from false surprise etc.), the boss wonders if they can catch cheats using only the sequence of throws. One of his lackeys argues: any sequence of throws is as probable as any other sequence (of the same length), so we can’t draw any conclusions. Sounds reasonable.

Now, while I’m no Brendon Brewer, I have been known to dabble in the dark arts of Bayesian probability. So let’s try to put this argument into mathematical form.

Let: = a series of n sixes is thrown by a particular player.
= the series 2, 1, 3, 5, 5, 6, 2, 1, 2, 6, … (a typical ordered sequence of n throws)
= background information (e.g. a die has 6 sides)
= the die thrown is fair and unbiased
= the die thrown is loaded, rigged to throw a six on cue (i.e. ). Once again, Ocean’s Thirteen furnishes an example. We will assume that the cheating player, having been smart enough to invent an undetectable loaded die, is dumb enough to use it indiscriminately, continuing to throw sixes without thinking that it might raise suspicions.

Then, the fact that the lackey refers to is this:

From which he concludes that:

i.e. a sequence of sixes, no matter how long, doesn’t make it any more likely that the die is loaded.

Having recast the claim in probabilistic terms, we can see that the claim is patently false. Bayes theorem allows us to express the probability that the die is fair given that n sixes have been thrown by a particular player:

It will help to write, using the law of total probability:

Where = “not ” = the die is not fair. We will make the simplification that if the die is not fair, then it is loaded i.e. . Then:

We’ve evaluated some terms above:, .

Now, we need the all important prior: . Note that . To consider a concrete example, suppose that the casino boss has received a tip-off that a player could use a loaded die that will slip past conventional security measures. Unfortunately, 10,000 players will be at the tables in casinos across the country tonight, and the cheat could appear at any one of them. Hence in this simplified case, the prior probability that a particular player is using a loaded die, before a single die is thrown, is at most:

Thus, we have;

Now, the all-important question is: how many consecutive sixes need to be thrown before it is more likely that the die is loaded than fair. The critical value is . Then, solving for n gives:

Thus, 6 sixes in a row will make it more likely that not that the player is cheating. If they want to be 99% sure that the player is cheating ( = 1%), then they will have to wait for 8 sixes.

Note carefully the moral of the story: so long as (it’s possible that a player is cheating), there is always some number of consecutive sixes that that makes it probable that the player is cheating. Note also the converse: no matter how many sixes a player throws, we can dispense with the loaded die hypothesis so long as a fair die is used by lots of people, lowering the prior. Both of these conclusions show that the sequence of sixes does call out for an explanation. The fact that a sequence of sixes is just as probable as any other sequence is irrelevant: what matters is that a sequence of sixes supports the hypothesis that the die is not fair, or that many, many people are actually playing.

In Leslie’s words:

“A chief reason for thinking that something stands in special need of explanation is that we actually glimpse some tidy way in which it might be explained.”

In other words, the hypothesis of a loaded die suggests both an explanation, and the need for one. Thus, we are indeed entitled to draw conclusions from the fine-tuning of the universe, because we can glimpse tidy explanations. Leslie’s primary conclusion regarding the fine-tuning of the universe is the same moral we have drawn from the loaded die fable: either many, many dice are being rolled (multiple universes plus the observational effect that universes that don’t permit observers cannot be observed), or the die has been manipulated by an intentional agent (guess who?). In other words, there is a selection effect at work: either an observational selection effect that “chooses” between actual universes, or an intentional selection in the mind of an agent who chooses between possible universes to find observer-permitting ones. (Or, to be completely rigorous, both. Leslie rightly rejects the possibility that “only one kind of world is logically or mathematically or cognitively possible”.)

So we can indeed draw conclusions from the fine-tuning of the universe. And what monumental conclusions they are!

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More of my posts on fine-tuning are here.

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