The sample mean is not relevant, you are not using a likelihood for estimating a population parameter. The interpretation you are using for the likelihood is not correct.

Routes available to evolution are not part of your calculation, even to name the particular pathway you claim is improbable.
You further presume that all steps are “not selectable” with no justification. You do not consider the probability of “scaffolding” that is known to produce IC structures. You do not consider sexual mixing, which can combine sequences in ways that bypass “non-selectable” steps.

Most of these calculations are not possible outside of special circumstances such as phylogenetic analysis. This is why we don’t see biologists using this approach at all. ID researchers have no special knowledge of mathematics that could make this possible. I am not the first to make these criticisms.

Vision can be of existential importance, and so selection is highly operative. There is plenty of intermediate functionality, so while the camera eye is complex it is not irreducibly so. Form follows function, and function responds to environmental pressure. What would be the barrier to convergent evolution of the camera eye?

First off, cephalopods don’t have the vertebrate camera eye; they have a distinct cephalopod camera eye. Second off, Nilsson D., Pelger S. A pessimistic estimate of the time required for an eye to evolve. Proceedings of the Royal Society of London, Series B 1994; 256:53-58.

Oh, I’d like to point out that some arthropods (and members of several other phyla too) also have a form of camera eye, and some cephalopods have camera eyes that lack some of the components of the canonical camera eye.

ID methods are unique in that the probability of an event-type is considered to be smaller the more often it is observed. Normally we think that events which occur more often are more likely, not less.

This is definitely a definition for design, you may say it doesn’t work, but it’s still a definition. But unselected steps are indeed more difficult to get through, and the more such steps there are in a row, the exponentially more difficult it is for evolution to get through them. Let’s say each step is probability .1 to get through, then two such steps would be .01 probability, and three such steps would be .001 probability, etc.

You are painting the bulls eye around the bullet hole. If we are talking about evolution, you would be calculating the probability of drawing a better hand than the other people at the table.

That’s the Sharpshooter fallacy. What you are ignoring is all of the beneficial interactions that didn’t happen. Using the same approach, you would be amazed that there are winners in so many lotteries because the chances of winning are 1 in 100’s of millions. What you would be ignoring is all the people that lost.

In the same way, there are many, many neutral mutations that have accumulated in any given genome. There may be many, many chances for a new mutation to interact with one of those accumulated neutral mutations and produce a beneficial phenotype. It is incorrect to single out the one beneficial phenotype that did evolve and calculate just the probability of that single phenotype.

Yes, I am, I am using the maximum likelihood value for estimating the mean, which is the sample mean.

No routes were excluded, two routes have been identified, and it’s reasonable to calculate the probability of resistance based on the rate at which it arises, 1 in 10^20.

The justification is that the chloroquine resistance rate is about the square of the atovaquone resistance rate. Evolution was not restricted in the interval of interest, scaffolding and any other path was available during this time.

Chloroquine resistance is not representative of evolution in general. Only very rarely are specific adaptations limited to just 1 or 2 mutations in the entire genome. Just looking at life in general will tell you there are many, many potential pathways for evolution to take, not just 1 or 2.

The response obtained in each generation would then be R = 0.00005m, which means that the small variation and weak selection cause a change of only 0.005 % per generation. The number of generations, n, for the whole sequence is then given by 1.00005^n = 80 129 540…

But isn’t this assuming that the selectable variation per generation becomes fixed in each generation? But with a generation time of only 1 year, I don’t think there is enough time for this to happen. I would propose then the following:

So that each time, the variation for the next step occurs within the fraction of the population that varied in the current step.

Finally, as they admit, the neural processing required for the more complex eyes was completely left out! This is a crucial factor, and would substantially reduce the probability.

Which makes it more surprising if evolution did it…

The probability of evolution as a cause gets smaller the more we see convergent structures. Let’s say the probability of an eye evolving is .1. Then the probability of it evolving twice is .1^2, and so on. It gets smaller, exponentially smaller.

No, you can indeed compute the probability of drawing a full house, before the fact! Before the bullet is fired.

That’s one probability. Another would be the probability of evolution generating a given structure.

Are you saying we can’t observe what evolution does, and compute a rate from that?

Now you are confusing the probability of an event viewed before the fact (1 in 100’s of millions) with the probability of an event after the fact (some people won). I’m not ignoring all the people that lost, that is why the probability (viewed before the fact) is 1 in 100’s of millions.

But Behe looks at what evolution did, including neutral mutations and existing variation.

But Behe is interested in the limits of what evolution can do, so he starts with a rare event (chloroquine resistance), and calculates a rate from there. This is a valid observation, regardless of what evolution can do elsewhere (Behe even has a chapter entitled “What Evolution Can Do”).

I don’t think you understand the terminology. The sample mean (a statistic) is an estimator of some population parameter. For a Normal distribution this estimates \mu, the population mean (and you also need to estimate the variance). You haven’t stated what it is you are trying to estimate or what the underlying distribution should be.

Example: To estimate \mu \text{ and } \sigma^2 for a normal distribution, you use the sample mean and variance to calculate the join probability (Likelihood) of all tions. and choose \mu \text{ and } \sigma^2 to Maximize the Likelihood. It turns out that the sample mean and variance ARE the maximum likelihood estimates of \mu \text{ and } \sigma^2, but MLE is a more flexible method of estimating parameters than Method of Moments (which is what people learn in an intro stats class). The sample mean is a Maximum Likelihood Estimator - it is not a likelihood.

But I digress. My not sure what you are trying to do, but you probably want a proportion or a rate, not the mean.

If the events were independent, and there were a multitude of ways to accomplish the same function, that might be so, but those are not reasonable assumptions. Various forms of eyes are thought to have evolved independently at least 40 times. Even algae can detect light and move towards it. Vision is clearly (heh) offers a big fitness advantage, or it would not have evolved so many times.

Response to light is ubiquitous in biomolecules. Even flowers face the sun; they do not have eyes because where are they going to run? For mobile surface and aquatic animals awash in light, any variation yielding competitive advantage in utilizing that environmental awareness would be subject to positive selection - in the land of the blind, the one-eyed man will be king. Populations which inhabit blackened situations such as caves often lose their sight, but to my knowledge this never happens in lit environments, demonstrating that vision is highly selective. The eye evolving in the first place is probably inevitable, and as the Nilsson Pelger study indicates, is likely rapid.

There is a selection of optically transparent crystallin proteins to choose from. Crystallin biomolecules serve other biological purposes as well. A refractive index is an inherent property of transparent materials, so a lens is entirely plausible by evolutionary mechanisms.

Here is the complete passage from the review (White, N. J. 2004. Antimalarial drug resistance. J. Clin. Invest. 113:1084-92) Behe pulled the number from:

“Chloroquine resistance in P. falciparum may be multigenic and is initially conferred by mutations in a gene encoding a transporter (PfCRT) (13). In the presence of PfCRT mutations, mutations in a second transporter (PfMDR1) modulate the level of resistance in vitro, but the role of PfMDR1 mutations in determining the therapeutic response following chloroquine treatment remains unclear (13). At least one other as-yet unidentified gene is thought to be involved. Resistance to chloroquine in P. falciparum has arisen spontaneously less than ten times in the past fifty years (14). This suggests that the per-parasite probability of developing resistance de novo is on the order of 1 in 10^20 parasite multiplications.“