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enter image description here Well,I'm supposed to find out the number of pink flowers to the red flowers and I started 'manually' selfing the generations. This seemed to manually impossible and so one colleague suggested I use nPr and nCr to solve it.I have the basic idea of the notation but have no idea how to relate it to the question in hand. Any help,intuitive ,mathematical or complete is appreciated. Note:If you don't know about selfing and gametes check out the Punnet square

N. F. Taussig
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Koushal
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1 Answers1

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My understanding of biology is poor, but I think I understand this problem. We start with a flower with alleles $R$ and $r$, or red and white, and we self fertilize it, so it has a child. The child has an equal chance of inheriting either allele, so there is a $.5$ chance that it is $Rr$, a $.25$ chance it is $RR$, and a $.25$ chance it is $rr$. You repeat this self-fertilization process on each generation for six generations, and want to count the number that are $RR$ or $rr$.

To start the problem, just look at the first generation, for which we described the chances of getting each (multi)-combination. We know that the $25\%$ of the first generation that have $RR$ will certainly also have children that are $RR$, and their children will have $RR$, and so on. The same holds for the $25\%$ that are $rr$. Thus, just by looking at the first generation, we already know that at least $50\%$ will be $RR$ or $rr$. Thus, we look at the $50\%$ remaining that are $Rr$. Then in the second generation, the same probability distribution applies to the children of the $Rr$ flowers, so $50\%$ of the children of the $Rr$ flowers will be $RR$ or $rr$, and the other $50\%$ will be $Rr$. That means we add another $25\%$ to the number of $RR$ or $rr$ flowers (that is, $50\%$ of the $50\%$). The pattern is obvious from there; we will have $1/2+1/4+1/8+1/16+1/32+1/64$ of the flowers be $RR$ or $rr$ after six iterations of this, i.e. $98.4375\%$.

Kevin Long
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  • The way I read this question the first generation of offspring is produced by crossing a plant with genotype RR that produces the phenotype of red flowers with a plant with genotype rr that produces the phenotype of white flowers. All of the offspring of these parents have the genotype Rr and the phenotype pink flowers. – N. F. Taussig Nov 04 '18 at 18:48
  • @N.F.Taussig I think that's the same as how I read it, I just worded it as starting it with those offspring, which form the $F_1$ generation. You do bring up a good point- I looked at it probabilistically because I thought of it as starting with a single flower, but as you point out, it's more like they just have some arbitrary number of offspring in the first generation, all with $Rr$. My argument should still work, though. – Kevin Long Nov 04 '18 at 18:59
  • @Koushal I'm not sure what you're saying, but let me try to clarify my point. If an $RR$ or $rr$ appears in any generation, its eventual descendant in the last generation will be $RR$ or $rr$ respectively. Then the only way to have $Rr$ in the last generation is if we have $Rr$ in each generation. There is a $1/2$ chance to get $Rr$, so over six generations, there is a $1/2^6$ chance to get $Rr$ in the end. Then $1-1/2^6$ is the chance of getting $RR$ or $rr$, which is what I gave above. – Kevin Long Nov 05 '18 at 19:41