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Sunday 8 January 2017

Probably the Worst Argument in the World

It's a well-known fact that 97.84392% of all probabilities cited in arguments claim a precision not justified by the methods employed.

In this outing, I want to spend a bit of time looking at something that comes up an awful lot in apologetics, sometimes made up on the spot, sometimes quote-mined from reputable sources, always used in a manner that betrays a failure to understand the underlying logic. That something is, of course, probability. Here's a recent example.
Looking at that, it's easy to see why this is leapt upon by apologists. All other considerations aside, these are huge numbers, which they think lends weight to their arguments. With such huge numbers, surely the things they're talking about are impossible, right?

Before we get to that, let's look a little bit at what probabilities are, how they're derived, and what's required to generate them.

Probability mathematics is, in the minds of a mathematician or a scientist, a robust mathematical framework that deals with predicting variables. In the mind of an apologist, however, it takes on a whole new complexion, being little more than a branch of apologetics.

The first thing to note is that all probabilities must fall between zero and one. An event with a zero probability is not going to happen - with some caveats, which we'll circle back to later - and an event with a probability of one is statistically inevitable - again, with some caveats. Probabilities can be expressed as percentages or, more often, ratios, but they always represent some number between zero and one.

Further, and this is the bit that can make it tricky when dealing with large numbers of variables, all probabilities must sum to one. 

Among the major reasons that apologists like to cite these numbers, aside from the fact that they're impressively large numbers, is that very few of the people they're likely to attempt to impress with these numbers are going to be in a position to do anything to gainsay them, and this is the motivation for this article.

The first thing to do is to check the numbers, and this is no trivial matter. I've said before that I'm no mathematician, but I can at least fumble my way through these calculations to show how probabilities work.

Starting with the first number above, we should begin by calculating the probability for a single royal flush. To do this, we need the probability for each card, which is taken by the number of cards that can contribute divided by the number of cards available. There's a simple equation that deals with this:

\( P(E)={r\over n} \) where \(P(E)\) is the probability of our event, \(r\) is the range of outcomes satisfying our conditions and \(n\) is the range of possible outcomes. Thus, to get any qualifying card, the probability is the number of qualifying cards divided by the number of all available cards. The first card is one of twenty cards from an available fifty-two, because any of the cards in any suit from ten to ace will satisfy the conditions. The second card is more restricted because, once the first card has been selected, the other three suits are excluded from our satisfying conditions, and our deck is one card short, so the card is one of four from fifty-one cards. The third is one of three from fifty, the fourth is one of two from forty-nine, and the final card is one from forty-eight.

Now we multiply all those together to get the probability of drawing a royal flush.

\(20/52 \times 4/51 \times 3/50 \times 2/49 \times 1/48 = 1.539 \times 10^-6 \), which equates to a probability of \(1:649,740\).

Now, to get the probability of five royal flushes in a row, we need to take that number and raise it to the power of five. A bit more calcium carbonate in the air, and we get:

\( 649740^5 = 115,797,189,947,256,250,531,862,400,000 \). To convert that to a percentage, we divide 1 by that number to get a ratio, then multiply it by 100.

\(1/115,797,189,947,256,250,531,862,400,000 \times 100 = 8.6357881435247595067029089939491^{-28}\%\). 

Remove the exponent by putting 28 zeros at the beginning, and we can see that the calculation is indeed correct, and that the percentage is \(0.00000000000000000000000000086357881435247595067029089939491 \% \), although of course, as highlighted in the funny at the top of the page, there's really no need to cite this to so many significant figures. We could just round it off pretty much anywhere. The only reason to do that is to make the number look as humongous as possible to confound the reader and make countering it more difficult. We could just as easily say that the probability is \(8.64 \times 10^{-28}\%\) and leave it at that. It's still a huge number (actually a tiny one), but now it doesn't look quite as impressive on the page, and that's really the point of the whole exercise.

So, even though there are some numbers in there that look really huge, even to the point of being intimidating to a non-mathematician, the derivation of them is really quite straightforward once you know how.

I'm not going to derive all the numbers in the graphic, not least because I don't want to make this post top-heavy with mathematics, and indeed I haven't checked them. I'd need to know where the numbers were obtained and how they were derived to mount any sort of serious attack on them, so I'm going to proceed as if they're correct because, as we'll see, they're really not going to do what the creationist who erected the argument needs them to do. The reason for this is rooted in how probabilities work and, once the underlying fallacy is exposed, we should be well-equipped to counter any such numbers thrown at us without having any need to be able to check whether they're correct.

As we've already discussed, probabilities must fall between zero and one. Generally speaking, any event with a zero probability can be thought of as impossible, but there are a few caveats to be noted here. For example, if you were to randomly choose any number on the real number line the probability of choosing any individual number is zero. That sounds counter-intuitive, and it is, and highlights one of the many problems one can run into when treating infinity as a number. The problem arises because the reals are infinite, and any finite number divided by infinity is zero. Because of this, any choice of number from the reals constitutes such a division, meaning that choosing, say, the number one, has a probability of zero, yet it's perfectly possible to choose that number at random from the reals.

This should be highlighting a serious point and, if there's one thing to take away from this post, that would be it, namely that events with a non-zero probability are perfectly possible. This is no mathematical trick, it's simply logic and integers.

I want to move now to a subtle point that's often lost on the apologist, and it's important for us to bear it in mind when we encounter these spurious probability calculations. It's something that my good friend the Blue Flutterby, the inimitable Calilasseia, highlighted in one of the great forum posts of all time, in my opinion. He quite rightly termed it the 'serial trials' fallacy. It's an extremely common feature of such arguments and, quelle surprise, the above commits it beautifully. A part of me would have liked to have simply reproduced Cali's post here as a guest post, but I'll link to it at the bottom while we look at some much simpler examples of the reasoning in action.

Let's take a fair die*. We're going to work out the probability of rolling all six numbers in a row. We'll keep it simple and allow them to come up in random order, so they don't have to come up in sequence, because that would make the numbers considerably larger, which would defeat our purpose here. Note that I'm flying somewhat by the seat of my pants so, if it doesn't work out in a simple manner, you'll never know that I used this example, because I'll have to start again, as simplicity is our benchmark for this exercise. Thankfully, we know we're only dealing with numbers from one to six, so we should be OK.

Let's work out what the probabilities are. As above, we only need work with \( P(E)={r\over n} \), except that, in this case, the number of available options doesn't reduce from throw to throw because, unlike the cards, all the numbers are available for every throw. This means that our probability calculation works as follows:

\(1 \times 5/6 \times 4/6 \times 3/6 \times 2/6 \times 1/6=0.015\), or \(1.5 \%\). Given this percentage, if we run a trial of this nature with 67 people, we would expect that at least one of them will get this result on the first attempt!

This is a simple example of a much broader principle. It doesn't matter what the probability is as long as it isn't zero because, if the sample set is large enough, the probability of any occurrence tends to one.

Now, looking at those numbers in the graphic, we can begin to see why they're complete nonsense, even if they happen to be correct. I'm going to crib a bit from Cali's work here, and simply point out that, even only working with the top 100m of seawater we can expect the number of interacting particles to be approximately \(3.07 \times 10^{43} \) particles, which starts to make that second number look a bit smaller. 

Another faux pas is to consider these probabilities in isolation. A really useful way to look at this is to consider a lottery. The odds of any particular person winning the UK lottery is approximately 1 in 14 million. That's a pretty low probability in the scheme of things. However, and this is the bit about how probabilities work that apologists seem to miss, and it's critical to defeating the above argument. The probability that somebody, somewhere will win the lottery at some point is exactly one. It's inevitable. There is literally no chance that the lottery will continue to go unclaimed. The way the UK lottery is set up, if there are four consecutive rollovers, the fourth jackpot is divided between those with 5 numbers. However, even if this were not the case, the jackpot would still be won with a probability of precisely one. 

There's one more thing we need to consider, and that's the idea of 'chance', as erected in the graphic. This is a canard that comes up with a regularity almost comparable to those caesium clocks we discussed in several past posts. Science doesn't posit 'chance', it posits well-defined natural mechanisms that can be quantified and predicted. In Has Evolution Been Proven? we looked at the distinction between a random system and one that is stochastic. A stochastic system is one whose future states depend on initial conditions plus one or more random variables, where random means 'statistically independent'. We looked at a system involving coins and variation. We saw there that evolution, properly described, is a stochastic system. Here, we're talking about exactly the same kind of system. We're not relying on a functional sequence of amino acids arising from scratch via chance, but via interaction upon interaction, with each new reaction being dependent on the compounds in existence plus one or more random variables. In short, each new state of the system brings down that probability, as each successive stage is built upon previous stages.

Finally, that last number, which is most definitely extracted directly from somebody's rectal sphincter. There is no way to derive a probability calculation for such an event without a designer, for the simple reason that a designer is a variable in the calculations, and we have no probabilities for a designer to work from. For that, we'd need some designers. Thus, this isn't a valid variable.

ETA: One thing I'd meant to include and overlooked during writing is a peculiar thing about how probabilities work that isn't immediately obvious. We've seen that, given enough time and/or a sufficiently large sample set, we can reasonably expect any event with a non-zero probability to manifest. What's not so clear is that even events whose probability is such that a vast swathe of time and/or a huge amount of resources in terms of interacting agents would be required in order to have them happen with a reasonable degree of expectation can happen immediately with a small number of interacting agents. We've looked in previous posts at, for example, radioisotope decay, and learned that this is a truly random process, with decay for a single atom happening at any time from straight away until the end of the universe. This same principle applies to all events that have non-zero probability. If an the probability of an event is 1 per 1,000 years, what this means is not that it will take 1,000 years to happen, but that it's statistically unlikely to happen more often than that, and even that probability has to be averaged out. Something with such a probability could happen twice in rapid succession and then not occur again for 2,500 years, which renders exactly the same probability.

In summary, if you encounter assertions containing probability calculations like the one above, step on them, because they're almost certainly bollocks.

Thanks for reading.

Edited to add: My friend Virphen, on reading this article, made the following incisive remark about the graphic at the top, which I felt was worth including:
Something that struck me is that if that meme just followed through on it's own initial logic, it could actually make something of a point.
Yes, if we got 5 royal flushes in a row we'd be pretty confident the game is rigged.
And when we get the amino acids... well we can be pretty confident the game is rigged there too - by selection.
Special thanks to @DoubleDumas for checking my numbers.

Calilasseia on the Serial Trials Fallacy

* It's a fairly common error to treat 'dice' as a mass noun, but it isn't. Dice is plural and die is singular.
We've simplified here, as 6/6=1
‡ Technically, it's undefined, which is slightly different, but this is a subtle technical point. We can say, however, that the result asymptotically approaches zero, which is sufficiently close to correct for government work. It's also extremely simplified because, in reality, when undertaking an exercise such as choosing a number at random, there are many factors impacting our choice. In all likelihood, if you're asked to do this, you'll choose a fairly low number, possibly one that has a special significance for you. However, it's still a single choice from an infinite set. These are both complications that don't appreciably impact the underlying point.

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