What will be the percentage of plants with red and white flowers in the final population?

Question

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

Answer 1

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. $98.4375\%$.