# Letters to Nature

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## Are The Dice Loaded?

July 28, 2008 by lukebarnes

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 $6^{10}$ 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: $S_n$ = a series of n sixes is thrown by a particular player.
$R_n$ = the series 2, 1, 3, 5, 5, 6, 2, 1, 2, 6, … (a typical ordered sequence of n throws)
$B$ = background information (e.g. a die has 6 sides)
$F$ = the die thrown is fair and unbiased
$L$ = the die thrown is loaded, rigged to throw a six on cue (i.e. $P(S_n | L\&B) = 1$). 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:

$P(S_n | F\&B) = P(R_n | F\&B) = 1 / 6^n$

From which he concludes that:

$P(L | S_n\&B) = P(L | B)$

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:

$P(F | S_n\&B) = \frac{P(S_n | F\&B) P(F | B)} {P(S_n | B)}$

It will help to write, using the law of total probability:
$P(S_n | B) = P(S_n | F\&B) P(F | B) + P(S_n | \bar{F}\&B) P(\bar{F} | B)$

Where $\bar{F}$ = “not $F$” = the die is not fair. We will make the simplification that if the die is not fair, then it is loaded i.e. $\bar{F} = L$. Then:

$P(F | S_n\&B) = \frac{P(S_n | F\&B) P(F | B)} {P(S_n | F\&B) P(F | B) + P(S_n | L\&B) P(L | B)}$

We’ve evaluated some terms above:$P(S_n | F\&B) = 1 / 6^n$, $P(S_n | L\&B) = 1$.

Now, we need the all important prior: $P(F | B) = p$. Note that $P(L | B) = 1 - p$. 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:

$P(L | B) = 1-p = \frac{1} {10,000}$

Thus, we have;
$P(F | S_n\&B) = \frac{6^{-n} p} {6^{-n} p + (1-p)}$

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 $P(F | S_n\&B) = 1/2$. Then, solving for n gives:

$n = \log_6 \frac{p} {1-p} = 5.14$

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 ($P(F | S_n\&B)$ = 1%), then they will have to wait for 8 sixes.

Note carefully the moral of the story: so long as $p < 1$ (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!

More of my posts on fine-tuning are here.

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Posted in Physics, Science, The Universe | Tagged anthropic principle, fine tuning, probability, universe | 10 Comments

### 10 Responses

1. That’s right (except that last sentence ). What matters is how the probability of what was actually observed varies *as a function of the hypothesis*. All of the other probability (of data sets that were not observed) could be redistributed around the data space arbitrarily (and in different ways for each hypothesis) and your inference remains exactly the same.

This is called the likelihood principle, and Bayesian Inference always satisfies it.

2. on July 30, 2008 at 8:02 am | Reply Owen

um… how many combinations are there of ten tosses of a coin? Or perhaps you meant rolls of a dice.

3. on July 30, 2008 at 9:06 am | Reply lukebarnes

Well spotted. I’ve corrected the quote

4. on July 31, 2008 at 12:25 pm | Reply dougaj4

Suppose in the casino analogy, we were just provided with the information that someone had thrown 10 sixes in a row, and no other information.

Possible explanations include:
There have been a huge number of people throwing dice for a very long time (and only information about 10 sixes in a row is transmitted).
The dice are weighted.
The dice were arranged that way.

Is this a reasonable analogy for the information we have about the physical constants of his Universe?

If so, it doesn’t tell us very much at all does it?

5. Hi Doug

I think that’s an excellent analogy and your conclusion is correct, that it doesn’t tell us much. But the reason for that conclusion isn’t due to the following fallacy:

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.

By the way, if anyone really wants to understand the anthropic principle, and how not to abuse it, I consider the following article to be a prerequisite.

http://www.cs.toronto.edu/~radford/ftp/anth.pdf

6. on August 3, 2008 at 2:17 pm | Reply lukebarnes

Brendon and Doug don’t seem to be very impressed by my conclusion, though they do agree. Let’s look at Doug’s options …

1. Many, many players with fair dice.
2. Weighted dice
3. The dice were arranged that way

I think options 2 and 3 collapse to the same idea: the hypothesis that the outcome of the throw of the die was manipulated by an intentional agent. The only difference is that a weighted die could fool someone into thinking it was a fair die, whereas just placing the die isn’t going to fool anyone. So were left with the loaded (a.k.a. manipulated) dice hypothesis or the many games of dice hypothesis.

In the context of the universe, this means that at least one of the following is true:
1.) There exists an ensemble of many, many (small-u) universes. This could be either large, causally disconnected spatial regions (bubbles), or previous cycles of an oscillating universe, or some other option. The first half of the 20th century taught us that the Earth, indeed the Milky way, is just a grain of sand in the vast expanse of the visible universe. Cosmic fine-tuning could tell us that the visible universe itself may be just a speck in the unimaginable size and diversity of the real universe. The constants that we all hold dear (gravitational, fine structure, properties of elementary particles, etc) may just be the result of random symmetry breaking. The search for the fundamental laws of nature may become the search for life-permitting options within an overwhelmingly lifeless landscape. Moreover, these other universes will presumably need to be generated by some mechanism, subject to some meta-laws of nature whose offspring are the laws of nature as we know them. (The alternative, that these other universes “just exist”, as a brute fact, seems unacceptable). But do these meta-laws need fine-tuning? It seems that if we’re searching for the ultimate laws of nature, cosmic fine-tuning could provide some vital clues.

2.) The life-permitting properties of our universe are the result of intentional causes. It would seem then that the fine-tuning of the universe, coupled with evidence against the existence of a universe-ensemble, furnishes a plausible design argument for the existence of God; one that, unlike William Paley’s, is immune from the effects of Darwinism. (Dallas Willard: “any sort of evolution of order of any kind will always presuppose pre-existing order and pre-existing entities governed by it. It follows as a simple matter of logic that not all order evolved”.)

At least one of these options has to be true, if Leslie’s reasoning is correct. Are these honestly unimpressive? Perhaps apatheism is more widespread than I thought.

7. on August 4, 2008 at 7:04 am | Reply dougaj4

“I think options 2 and 3 collapse to the same idea: the hypothesis that the outcome of the throw of the die was manipulated by an intentional agent.”

No, I have two important points of difference.

Option 2 was not supposed to be an analogy for an intelligent agent providing wheighted dice (although it could be of course, but then it would be just like 3, as you said), the intended analogy was that the fundamental constants are what they are because that is what they have to be (for some reason unknown to us).

Secondly the dice being arranged does not necessarilly imply an intelligent entity intentionaly arranging that way. They may have been arranged by some unintelligent process which we don’t know about. Even if they were arranged by some intelligent entity, I think it is misleading to call this an analogy for a god figure, because the entity may not have any of the features that “gods” traditionally have.

Brendon – I’m reading that paper, but I won’t comment until I’ve finished it.

8. [...] that talk, I read some internet articles that were rather woeful. It’s time to quote John Leslie again: “The ways in which ‘anthropic’ reasoning can be misunderstood form a long and dreary [...]

9. [...] and gets four aces each time – call this “M”, the “magic deal”. The probability of M, assuming he is dealing fairly, is approximately p(M | fair-deal) = one chance in . Now suppose that there are other poker games [...]

10. on January 5, 2013 at 7:19 am | Reply James

The problem with equating drawing a 6 on a die to *Complexity is obvious. But atheists are not using reason but bias.

We are not asking for something to happen that is no different than a 4 is to a 6. We are asking for extreme complexity to assemble itself when garbage should be the norm. A 6 is no different than a 4 in this respect. However consciousness is much different than a ball of crap. One can comprehend the reality it is in–see it when it cannot know there even something to see, smell it touch it, taste it, and hear it.

So all theists are showing is a darkened intellect and denial of the obvious in these infantile demonstrations.

You have to come to the conclusion that is actually something wrong with these people. That in denying God and constantly attacking those who clearly perceive Him they have corrupted their thinking beyond repair.

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