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Then to compute the probability that mutation A will not occur in G generations (call it nGA), we again use the multiplication rule and obtain:
P(Ac) = ((1 − P(BeneficialA)𝜇)^n)^nGA = (1 − P(BeneficialA)𝜇)^n*nGA
If G is the number of generations, why don't you just write P(Ac) = ((1 − P(BeneficialA)𝜇)^n)^G = (1 − P(BeneficialA)𝜇)^n*G ? Why call it nGA instead?

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Good, I'm glad you are paying attention. I could have simply labeled the number of generations G but you have to keep track of generations for different steps in the evolutionary process. n and nGA are variables but the product of these two numbers is simply the total number of replication trials use to compute the probability that mutation A occurs.However, once mutation A occurs, that member becomes the progenitor for a new lineage which are candidates for mutation B. The population size n is not used for computing the probability of mutation B occurring on some member with mutation A, that population size will be nA, the number of members with mutation A and the generations for this part of the computation will be nGB, the numbers of generations members with mutation A are replicating. This is where amplification becomes critical. If members with mutations A can't either increase in number and or replicate for many generations, you will not have enough trials for there to be a reasonable probability for mutation B to occur.The probability equation for the mutation B is written in the same manner as for mutation A but with different population size and number of generations of replication. I'm sure you can easily do it. And how do compute the joint probability of the two probabilities? I'm sure you know that as well. If you don't want to do it, I'll post the equations.

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The mathematics is self evident. Again, we'll stop at this point for questions, comments, complaints...

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So far, it's all been exceptionally self-evident, I'll grant you that.

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Question: when do we get to the dinosaurs?

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In case you don't recognize it, these probability equations are the general solution for rmns and apply to all replicators. I'll show you how to do the calculation for multiple simultaneous selection pressures after we finish with the single selection pressure model.

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In case you don't recognize it, these probability equations are the general solution for rmns and apply to all replicators.

There's no reason to believe that at all.

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There is if you understand that rmns is a stochastic process and understand how to do probability calculations.

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However, once mutation A occurs, that member becomes the progenitor for a new lineage which are candidates for mutation B

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As I stated earlier it is often not the case that two beneficial mutations have to occur in the same lineage. Therefore any equation that assumes otherwise cannot represent the general case as you claim.

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The mathematics I'm presenting here is the mathematics of rmns by common descent. How do you think replicators accumulate the mutations necessary to adapt to selection pressures by rmns? Perhaps you think that lateral transfer of genetic material is the way it is done? Try doing the mathematics of random ecombination. If you can't do it, I'll show you.

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In case you don't recognize it, these probability equations are the general solution for rmns and apply to all replicators.

And indeed so far they apply with a small change in terminology to dice and playing cards. In order to apply them to dinosaurs, though, at some point we need to plug in some relevant numbers, such as the population size of the dinosaurs from which birds descended, the mutation rate of dinosaurs, etc.

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So, without wishing to hustle you I am interested to know when you're going to get on to the good bit.

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We are almost finished with the mathematics and then we'll do the arithmetic. And the mathematics and arithmetic apply to any replicator, dinosaurs included.

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Hi Kleinman,
Weinrich et. al. believe their research indicates that selection constrains evolution to narrower pathways than previously supposed, and that that makes evolution more predictable and repeatable than we might have expected. What is it about their data that leads you to instead conclude that evolution is impossible?

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Hi to you Percy. What I think Weinreich is saying is that any evolutionary trajectory must take a path of ever increasing fitness. If an evolutionary trajectory requires a detrimental mutation, amplification does not occur of that variant so the probability of another beneficial mutation occurring on a member of that lineage remains low. I actually think that Weinreich made an error in his paper when he said there were only 120 possible evolutionary trajectories. It doesn't say how he came up with that number but I suspect he is computing permutations (5!) which is n different things (5 mutations) taken all at a time. He should have used permutations of n different things (4 bases) taken k (5 mutations) at a time with repetition or 4^5=1024 possible pathways. But even that might be a low estimate because you might have 6 mutation variants and so on. Weinreich did publish a follow-up paper where he identified more resistant variants.And don't get me wrong, I'm not saying that evolution is impossible, I'm giving you the mathematical rules which govern how evolution by rmns works. It is the theory of evolution which is mathematically irrational based on how rmns works. It is the multiplication rule of probabilities which kills the theory of evolution.

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I'd like to suggest, strongly, that it isn't necessary to explain simple math and probability in painful detail. You've gone on for almost as long as the Constitution. Time to get to the crux.

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The painful part is already over. The calculation for computing the probability of a B mutation on a member which already has the A mutation is done in the exact same manner as computing the probability of the A mutation occurring on some member of the entire population. The only thing that changes is you don't use the entire population size, you only use the portion of the population with mutation A. So if that subpopulation with mutation A does not increase in size, the probability of mutation B occurring on some member with mutation A will be small. That is because the joint probability of mutation B occurring on some member with mutation A is computed using the multiplication rule.

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The mathematics I'm presenting here is the mathematics of rmns by common descent. How do you think replicators accumulate the mutations necessary to adapt to selection pressures by rmns?

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Sexually reproducing creatures - like dinosaurs - receive genetic material from both parents. Thus it is possible to inherit advantageous mutations from both parents.

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That's right. So then how does this change the mathematics? Instead of a haploid replicator, you now have a diploid (ignoring polyploid replicators for the moment). What this effectively does is double the initial population of genomes replicated per generation. Instead of n replicating haploid genomes, that number becomes 2n because of diploid. And then you can try to include recombination into the calculation. I've done the recombination calculation alone to study why recombination does not have a significant effect on the evolution of drug-resistant HIV. The empirical evidence already is clear that rmns is not significantly altered by sexual reproduction. Combination herbicides are already known to impair the evolution of herbicide-resistant weeds. There are other examples as well.I suspect you would have to include the mathematics of Mendelian genetics as well. You should try and do the calculation.

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It is the multiplication rule of probabilities which kills the theory of evolution.

Ooh, good! This is something to look forward to.

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Though it seems odd that something so simple and almost universally known should overturn a scientific theory, and that no-one should have noticed it before. It would be one thing if some rare genius were to find a flaw in the theory, but if you are right about the multiplication rule, then you are not a rare genius --- you are a man of middling intellect who has achieved distinction by living among a species of complete morons who cannot apply the math they learned in middle school. Well ... they can.

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What I think happened is that evolutionary biologists went down a wrong track when the got stuck on the notion of fixation rather than recognizing its amplification which affects the probabilities, not relative frequencies. Did you notice that relative frequencies don't appear in these probability equations (they do for random recombination)? You don't have to be the sharpest knife in the drawer to figure out this problem. rmns is nothing more than a nested binomial probability problem where the different binomial probabilities problems are linked by the multiplication rule of probabilities. If you did a careful study of a good computer simulation of rmns like Tom Schneider's EV computer simulation and looked carefully at the empirical examples of rmns, its not that hard to see how rmns works.

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Kleinman writes:

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I actually think that Weinreich made an error in his paper when he said there were only 120 possible evolutionary trajectories. It doesn't say how he came up with that number but I suspect he is computing permutations (5!) which is n different things (5 mutations) taken all at a time. He should have used permutations of n different things (4 bases) taken k (5 mutations) at a time with repetition or 4^5=1024 possible pathways.

See Figure 2 of Weinrich et. al.. The specific mutations are known, he's just stating the number of possible orderings to arrive at the final TEM state, which is 5!.

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Ok, and I re-read the paper and found where he explicitly does the 5! calculation. It really doesn't matter from the point of view of computing the mathematics of rmns. Each lineage on any particular evolutionary trajectory must still solve nested binomial probability problems linked by the multiplication rule of probabilities. And the only way a lineage can do this efficiently is by amplifying the current beneficial mutation to improve the probability of the next beneficial mutation occurring on one of its members. If the amplification process does not occur, the probabilities of the next beneficial mutation remain low.

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And don't get me wrong, I'm not saying that evolution is impossible, I'm giving you the mathematical rules which govern how evolution by rmns works. It is the theory of evolution which is mathematically irrational based on how rmns works. It is the multiplication rule of probabilities which kills the theory of evolution.

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Well, if that's it then we'll just have to see if anyone wants to engage with you further about whether RM/NS renders evolution irrational.

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Until then, please move on to addressing the dinosaur-to-bird issue. Sorry to be curt, but you've pretty much exhausted everyone's patience.

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It's not rmns that renders evolution irrational, rmns works in a mathematically rational way where each evolutionary step is governed by the multiplication rule of probabilities. rmns is nothing more than a set of nested binomial probability problems where the individual binomial probability problems are linked by the multiplication rule of probabilities. This is why the only examples of rmns that work efficiently are those with single targeted selection pressures. When a population is subjected to more than a single targeted selection pressure simultaneously, the ability of variants to amplify any particular beneficial mutation for one selection pressure is impaired by the other selection pressures. Try to find an empirical example that doesn't demonstrate this. They don't exist.So the reason why dinosaurs can not transform alleles which produce scales into alleles which produce feathers by rmns is that too many genes must be transformed.Now perhaps in your mind there exist a set of targeted selection pressures, targeting each appropriate gene that would transform a scale into a feather and this set of selection pressures somehow occur consecutively so that one selection pressure does not interfere with the others, then you would have a mathematically valid argument that rmns could transform scales into feathers. But you don't have these selection pressures. And any selection pressure that targets more than a single gene at a time interferes with the rmns process. This is why combination therapy works for HIV, this is why combination herbicides suppress the evolution of herbicide-resistant weeds, why every real, measurable and repeatable example of rmns demonstrates this. It is all due to the effect of the multiplication rule of probabilities.

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So the reason why dinosaurs can not transform alleles which produce scales into alleles which produce feathers by rmns is that too many genes must be transformed.

How many genes must be transformed to make this change?
What is the limit of the number of genes that could allow the transformation?

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I had a link (which is now dead) to a paper where biologists looked at this problem. What they did is looked at the genomes of birds and the genomes of reptiles and tried to determine which genes would need to be transformed to turn scales into feathers. They identified at least eight genes.Once you get above the transformation of a single gene by a single selection pressure, rmns is stifled. How much is it stifled? Consider the evolution of HIV where only 2 genes are targeted by 3 selection pressures and you have people surviving for decades instead of weeks. If you have huge populations such as seen with Malaria (trillions or more), 2 selection pressures still allows for the emergence of resistance. This happens because with populations this large, the probabilities of double beneficial mutations become realistic. But each evolutionary step means that double beneficial mutation variant must amplify into the trillions as well for the next set of double beneficial mutations. This is why 2 drug therapy will not be adequate for durable treatment of Malaria. 3 drug therapy will be required, in particular with those patients with impaired immune systems.

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What I think happened is that evolutionary biologists went down a wrong track when the got stuck on the notion of fixation rather than recognizing its amplification which affects the probabilities, not relative frequencies.

Well, fixation is just amplification taken to 100%. And it's easy to work with. It's likely to be a good approximation to math which took the steps from mutation to fixation into effect: such an approximation would start breaking down if and to the extent that beneficial mutations were so common that it's highly likely that a second beneficial mutation will arise before the first one's achieved fixation. Even so, this will not have much qualitative effect: it will still (for example) be the case that evolution goes at a higher rate when there are multiple (soft) selection pressures, because the reasons for that will still apply.

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Fixation is not the same as amplification. What if the total population size is 10? If you need a million replications to have a reasonable probability for a particular mutation, that would require 100,000 generations. And fixation is not necessary for amplification to occur. The relative frequencies of the particular variants evolving to a selection pressure can remain constant while the entire population size is increasing. You have to use the number of replications to determine your probabilities.

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You don't have to be the sharpest knife in the drawer to figure out this problem.

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That would be kinda my point.

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Every week, hundreds of people realize that they can overturn either evolution or the Big Bang with reference to some snippet of math or science they learned in middle school. So far, they have invariably been wrong; and it is easy to see why: if it could be done that easily, it would have been done already.

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I actually learned about the multiplication rule of probabilities in elementary school. Maybe evolutionists don't want to think about the multiplication rule for stochastic processes. It kinda gums up the theory of evolution. But it is very useful to understand this if you want to develop durable treatments for cancers and infectious diseases.

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If you did a careful study of a good computer simulation of rmns like Tom Schneider's EV computer simulation and looked carefully at the empirical examples of rmns, its not that hard to see how rmns works.

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A good computer simulation actually shows that my math in post #132 is correct. Which is nice for me.

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But there is a difference between fixation and amplification. Now random recombination is dependent on relative frequencies of alleles in a population. Now I haven't done the mathematics for rmns for sexually reproducing replicators including recombination. I also think you would have to include Mendelian probabilities in the calculation. I don't think this case will rescue the theory of evolution because the empirical evidence already shows that combination selection pressures stifles rmns (eg combination herbicides) for this class of replicators.

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Fixation is not the same as amplification.

Well, fixation is amplification to 100%

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Are you sure about that? No more replications after that?

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What if the total population size is 10? If you need a million replications to have a reasonable probability for a particular mutation, that would require 100,000 generations.

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And then fixation, if it occurred at all, would occur quickly, and very likely before any other given beneficial mutation arose. This would be a splendid example of a case where you could entirely neglect amplification as such and just think about fixation.

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Let's make the example even simpler than that. Let's say you have a population of 10 exact clonal matches. Every allele is already fixed. but you have only 10 replications. To get a million replications with a constant population size over the generations would require 100,000 generations

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I actually learned about the multiplication rule of probabilities in elementary school. Maybe evolutionists don't want to think about the multiplication rule for stochastic processes.

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The people who worked out the math of the theory of evolution think about the laws of probability quite a lot.

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Of course they do and the earliest paper I know of where the multiplication rule applies to rmns was discussed is Edward Tatum's 1958 Nobel Laureate Lecture. But Haldane and Kimura don't address this aspect of evolution in their models. I have a big advantage over Haldane and Kimura, they didn't have all the empirical evidence of rmns that is available today.

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But there is a difference between fixation and amplification.

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And a simulation, which has no particular concept of fixation, still produces the same qualitative results as math which uses the concept of fixation, because it is in fact a good approximation.

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You can have amplification without fixation simply by all variants increasing in number yet still maintaining the same relative frequencies and you can have fixation without amplification simply by having all other variants dying out.

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I don't know why the rest of your post is about recombination.

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Because the probabilities of random recombination are dependent on the relative frequencies of variants in the population, not the absolute number of each variant.

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I don't think this case will rescue the theory of evolution because the empirical evidence already shows that combination selection pressures stifles rmns (eg combination herbicides) for this class of replicators.

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Do you have any empirical evidence that this happens when the selection pressures are not hard? You know, cases where the selection pressures aren't us hitting the population with the most virulent poisons our ingenuity can devise?

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The mathematics of rmns is not dependent on the intensity of selection. The question is does the evolutionary trajectory (mutations required) to resistance to the selection pressure(s) change depending on the intensity of selection? The point I think you are trying to make is that if the intensity of the selection is low, amplification will be easier for the remaining variants.I posted what I thought was a very interesting video of the evolution of resistance of bacteria to an antibiotic. Here's the link to the video again:
Scientists create video of bacteria evolving drug resistance.
I've written to the scientist who made this video and asked that he repeat the experiment with 2 and then with 3 drugs using the same technique of low to high-intensity selection with the combinations. That would give some empirical answer to this question. What I suspect is that if the bacteria are able to evolve resistance to the combinations, it will be much, much slower than to the single drug experiment.Watch the video and tell me what you think.

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So your answer to these questions

Theo writes:

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How many genes must be transformed to make this change?
What is the limit of the number of genes that could allow the transformation?

is?
I mean really they are simple questions. You have made it very clear that you have figured this all out therefore you must know the answers.
Sorry but that you have a dead link that asserts something doesn't cut it.
Is the answer for the first question 8 or not?
You know the math, are you saying you don't know the answer for the second question?

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The web site where this link was posted is a university web site. The web site is still there and I contacted the professor who originally posted the paper and asked for a copy but no response. I'm sure I still have the link and you can contact the professor yourself and see if you can get the paper. What I do remember is that they listed at least 8 genes necessary to be transformed.If you want, I'll search my older computer and find the link and perhaps you will have better luck finding the paper.

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First, let's look at the physics behind the story. The lift equations for rigid wings are straightforward enough. Bumble-bees are fairly big, weighing almost a gram, and have a wing area of about a square centimetre.Tot up all the figures and you find that bees cannot generate enough lift at their typical flying speed of about 1 ms.
But that doesn't prove that bees cannot fly. All it proves is that bees with smooth, rigid wings cannot glide, which you can show for yourself with a few dead bees and a little lacquer.

So here we have an example of math/physics at work, which shows for a certain set of parameters bumblebees can't fly!

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But bumblebees, not having read of this, continue to fly just fine.

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So, what does this tell us?

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If math and physics professionals model the wrong variables they get the wrong answers, even if all the math is correct.

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And, as often is the case, math and physics professionals usually know squat about biology and related subjects. (Increasing one's knowledge of math and physics does not correct this deficiency.)

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Early in my career, I worked in the aerospace industry. We had a saying "Put a big enough engine on anything and you can make it fly". I assure you that I have enough training in microbiology, biology, organic chemistry, biochemistry, genetics,... to understand rmns. You show me your degrees and I'll show you mine. And I'm pretty sure I've had a lot more training in mathematics and physics than you.

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Kleinman writes:

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And I'm pretty sure I've had a lot more training in mathematics and physics than you.

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And I'm sure you haven't. After all, you haven't answered any of my questions about mathematics:

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You have a standard deck of 52 cards. You randomly choose a card.

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What is the probability of having drawn the Ace of Spades?
What is the probability of having drawn an Ace?
What is the probability of having drawn a Spade?
What is the probability of having drawn a black card?
What is the probability of having drawn a card?

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What is the probability of having drawn the Ace of Spades given no information?
What is the probability of having drawn the Ace of Spades given that it is an Ace?
What is the probability of having drawn the Ace of Spades given that it is a Spade?
What is the probability of having drawn the Ace of Spades given that it is a black card?

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How boring! Let's do something fun. Let's do a card drawing problem that has a relationship to population genetics.You have a standard deck of 52 cards. You randomly choose 3 cards, 1 at a time with replacement. Write the probability function that you choose either a queen or an ace or king in the 3 draws.

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Suppose you have a single-gene trait with two alleles with perfect dominant/recessive expression. If you are homozygous for dominant allele or heterozygous, you display the dominant trait. Only if you are homozygous for recessive allele do you display the recessive trait and you always do if you are homozygous recessive. Suppose the current rate of recessive display is 1-in-1,000. Suppose that those who display recessive trait are sterile and cannot reproduce while those who display dominant trait (either homozygous dominant or heterozygous) have no difference in reproductive capability.

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How many generations would need to pass in order to reduce the appearance of recessive trait from 1-in-1,000 to 1-in-1,000,000?

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To help you start: What is the value for p? What is the value for q?

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That calculation is already done: http://pba.ucdavis.edu/files/149789.pdfLet's do a different problem. Let's think about how a population can evolve resistance against two selection pressures simultaneously.Start with a population size N. In a single generation of replication of that population with a given mutation rate, what is the probability that some members will get a beneficial mutation to one of the selection pressures, other members will get a beneficial mutation to the other selection pressure and what is the probability that some members will get beneficial mutations to both selection pressures?Since you like hints, I'll give you a hint. The members that get beneficial mutations to the first selection pressure are one subset of the population, the members which beneficial mutations to the second selection pressure are a second subset of the population and the members which get both beneficial mutations are the intersection of the two subsets.

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Kleinman writes:

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Early in my career, I worked in the aerospace industry

Hey David Coppedge, do you go by the name of Kleinman nowadays?
I mean, David Coppedge used to work as a computer specialist in that industry where he pretended to know more about biology than biologists and more about geology than geologists and more about maths than mathematicians and more about physics than physicists. Didn't bode too well for him in the end.

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Who is David Coppedge. The multiplication rule of probabilities does not bode well for the theory of evolution.

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Kleinman, are you a YEC?

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I already answered that question in a previous post. Pay attention.