Total Drek

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Monday, April 13, 2009

The funny thing about probabilities.

In a number of the classes I've taught I always make a point of talking for a bit about probability. This is partly because I just find probability and statistics cool as hell. Seriously, I'm that guy. Additionally, however, this is because I think a basic understanding of probability is very useful in a person's life. One way in which I think it is useful is helping to dispel nonsense ideas about how unusual or unbelievable an event is.

See, as many of you know by now many events that may seem unusual or noteworthy are- from the perspective of probabilities- quite unremarkable. A good example is to think about flipping a coin: unless we worry about it landing on precisely the edge, there's a 0.50 probability that it will land heads or tails. Moreover, these probabilities are independent, meaning that one flip does not influence the next flip.* Given these two facts, we can find the total likelihood of a series of flips by multiplying the probabilities of each flip together. What this means is that the probability of, say, three heads in a row is: p= 0.50*0.50*0.50= 0.125 (i.e. a 12.5% chance). Similarly, the probability of five heads in a row is: p= 0.50*0.50*0.50*0.50*0.50= 0.031 (i.e. 3.1% chance). Obviously, the more flips we add, the lower the probability of five heads in a row. Most people grasp this type of logic on a fundamental level.

The problem emerges because most people don't ask themselves the next question: what's the probability of, say, a heads then two tails, then two heads? Well, the probability of a heads or a tails is 0.50, so that calculation would look like this: p= 0.50*0.50*0.50*0.50*0.50= 0.031 (i.e. 3.1%). If you're feeling like you've seen this before, you have at least a marginally functional memory: it's the same probability as the five heads in a row. In fact, it doesn't matter which specific sequence of heads and tails we're interested in- we'll always get the same probability for any set of five flips.** And even more fun, while the likelihood of any one of those combinations is only 3.1%, the likelihood that we will get one of them when we flip a coin five times is 100%. In other words, while a guess as to the specific outcome will only be correct 3.1% of the time, nevertheless, we know with certainty that one of those low probability outcomes must result from the process of coin flipping. This is important because it teaches us an important lesson: just because an event is improbable, it does not mean that it is all that interesting. Put differently, any time you flip a coin five times in a row, there is a 100% chance that some improbable event (defined as p< .05) will occur. This is, as a side note, why researchers should not fetishize statistical significance. Significance is an extremely useful heuristic for dealing with the confirmation bias and related foolishness, but in and of itself cannot tell us when something is important.

This is also important because it relates to the arguments of Intelligent Design Creationists.*** One of their primary "arguments" is that life cannot have evolved because the cumulative probability of a sequence of mutations producing the structures we see is very small. As you might surmise at this point in the post, this is problematic at best because some outcome from any sequence of mutations is inevitable, even if any particular outcome is vanishingly unlikely. So, staking your argument on the idea that this set of outcomes is unlikely is implicitly assuming that this set of outcomes is the only way life could have worked. Add in the fact that natural selection and evolution work to make some things more likely, and you start to see the weakness of probability based arguments for design.

And this brings us to the real point of this post:**** it looks like those arguments have just gotten a bit weaker.

Researchers Paul Higgs and Ralph Purditz of McMaster University in Canada***** have shown that 10 of the 20 amino acids commonly in use in terrestrial biology are thermodynamically favored. What does this mean? Simply that the basic laws of physics favor the development of these ten amino acids and, moreover, their prevalence is predicted with a high degree of accuracy by those laws. In other words, folks, when it comes to half of the amino acids biology uses we aren't talking about a series of unrelated coin flips. We're talking about a series of flips with a very, very weighted coin. To quote the abstract:

Of the twenty amino acids used in proteins, ten were formed in Miller's atmospheric discharge experiments. The two other major proposed sources of prebiotic amino acid synthesis include formation in hydrothermal vents and delivery to Earth via meteorites. We combine observational and experimental data of amino acid frequencies formed by these diverse mechanisms and show that, regardless of the source, these ten early amino acids can be ranked in order of decreasing abundance in prebiotic contexts. This order can be predicted by thermodynamics. The relative abundances of the early amino acids were most likely reflected in the composition of the first proteins at the time the genetic code originated. The remaining amino acids were incorporated into proteins after pathways for their biochemical synthesis evolved. This is consistent with theories of the evolution of the genetic code by stepwise addition of new amino acids. These are hints that key aspects of early biochemistry may be universal. [emphasis added]

On a scientific level, this is a really neat finding. But on another level, it's one more nail in the coffin of the hoary old creationist probability argument. And that's just fun.

Albert Einstein is sometimes credited with remarking that god does not play dice with the universe. This most recent finding, though, suggests that if he does, the dice are loaded.

* When the outcome of one event influences the likelihood of a second event, these events are said to be conditional probabilities.

** If this seems intuitively wrong, it's for a simple reason: Most of us don't distinguish strongly between "three heads and two tails" and "three tails and two heads". These sets of outcomes tend to get lumped into a common category of "mixed heads and tails" in our minds. Since there are so many more mixed outcomes than non-mixed (obviously, there are only two possible non-mixed outcomes, the total likelihood of which is 6.2%, meaning that all mixed outcomes occupy the remaining 93.8% of the available probability space) this gives the impression that it is the non-mixed combinations that are so very unlikely. Nevertheless, each mixed outcome is a distinct result that is just as unlikely- on its own- as getting five heads.

*** C'mon, you knew we were getting here eventually!

**** What? You didn't think just felt like talking about math, did you?

***** Shout out to Tina of Scatterplot for the work of her colleagues! Woot!

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Blogger Marf said...

You can't totally write-off landing on edge as an impossibility. I have once, flipped a nickel, it landed on the table and began spinning, and came to rest on its edge.

The probability is extremely low. However with each successive flip, the possibility of it landing on its side at least once in the series becomes greater. (Same thing as saying there's a greater chance if I flip it 100 times vs. 2 times)

Even something as unlikely as a coin coming to rest on its edge, given enough flips, will happen at least once.

Even if those other 10 amino acids are as unlikely to form as a coin coming to rest on its edge... Give it a few billion years of coin flipping.

Monday, April 13, 2009 12:03:00 PM  
Blogger Drek said...

I wrote off landing on the edge not so much because it can't happen, as to simplify the example. You're right- the probability is low, but non-zero, so given enough trials it should occur eventually.

I suspect this just isn't coming across as you intend it, but literally speaking your statement that, "...with each successive flip, the possibility of it landing on its side at least once in the series becomes greater," isn't correct. Each time you flip the coin you have exactly the same probability of getting heads, tails, or the edge. What I think you mean is that as you increase the total number of flips, the aggregate probability of at least one of those flips being an edge rises, but this doesn't change the likelihood of it occurring on any particular flip. So, for example, if the likelihood of a coin landing on its edge is 0.0002 then in 100 flips we would expect it to occur 0.02 times or, in other words, it has a probability of 0.0002. In contrast, if we flip the coin 1000 times we would expect an edge landing 0.2 times, for a probability of 0.002. The overall probability is rising, but the likelihood of an edge landing on any flip remains the same.

Again, I think you know this given your parenthetical remark, but I wanted to make sure this was clear to others given that people frequently lapse into thinking that an event is "due" or somehow more likely because of a series of other events (e.g. "He's been striking out a lot lately so he's due for a home run").

Monday, April 13, 2009 12:24:00 PM  
Blogger Marf said...

Yes, you are correct. I just didn't know how to word it right.

Monday, April 13, 2009 12:32:00 PM  
Blogger tina said...

And you thought McMaster was just known for its cutting-edge sociology...

Monday, April 13, 2009 6:34:00 PM  
Anonymous Elizabeth said...

Thanks, I hadn't even heard about this finding, but it's very interesting.

Of course, the creationists, excuse me, intelligent design theorists, will just say that god designed the amino acids this way. When you press them, they always move around the goalposts.

Tuesday, April 14, 2009 10:16:00 PM  

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