The Shepherd's Farm

Around the Ladoum Sheep and More. A Blog from Jacques E. Boillat, Sunday April 17 2022

The \(F\) Coefficient

It is well known, that when you mate a male and a female, te outcome will get exactly half of the genes of the Father and half of the genes of the mother.

In particular, if you crossbreed a male of race A with a female of another race, half of the genes of the offspring will be A-genes. The female will not participate in adding any A-genes to the outcome.

For reason of simplicity, suppose that we repeat the mating many times, and that the outcome is always a female. That outcome will again be mated with a male of race A, etc.

Crossbreeding: The squares represent males, and the ellipses represent females in the figure. The numbers represent the purity coefficients of the sheep. The first generation child gets half of the genes of its mother, and half of the genes of its mother, i.e. \(\frac{1}{2} \times 0 + \frac{1}{2} \times 1 = \frac{1}{2}\). The second generation child gets half of the genes of its mother, and half of the genes of its mother, i.e. \(\frac{1}{2} \times \frac{1}{2} + \frac{1}{2} \times 1 = \frac{3}{4}\), etc.

We want to find a formula for the purity of the \(n^{\text{th}}\) generation child. Let call \(F_n\) the coefficient of purity of the outcome at the \(n^{\text{th}}\) generation. If \(n = 0\) we talk sometimes about the parent generation, and may also write \(P_0\). \[ \begin{align} F_0 &= 0 \\ F_1 &= \frac{1}{2} F_0 + \frac{1}{2} = 0 + \frac{1}{2} = \frac{1}{2} \\ F_2 &= \frac{1}{2} F_1 + \frac{1}{2} = \frac{1}{4} + \frac{1}{2} = \frac{3}{4} \\ F_3 &= \frac{1}{2} F_2 + \frac{1}{2} = \frac{3}{8} + \frac{1}{2} = \frac{7}{8} \\ F_4 &= \frac{1}{2} F_3 + \frac{1}{2} = \frac{7}{16} + \frac{1}{2} = \frac{15}{16} \\ \end{align} \] i.e. we guess that \[ F_n = \frac{2^{n}-1}{2^n} \] We can rewrite it as \[ \begin{align} F_0 &= 0 \\ F_1 &= \frac{1}{2} = 1 - \frac{1}{2} \\ F_2 &= \frac{3}{4} = 1 - \frac{1}{4} \\ F_3 &= \frac{7}{8} = 1 - \frac{1}{8} \\ F_4 &= \frac{15}{16} = 1 - \frac{1}{16} \\ \end{align} \] i.e. we guess that The degree of purity of an offspring of the \(n^{\text{th}}\) generation is \[ F_n = 1 - \frac{1}{2^n} \]

Here the proof. For mathematics friends only!

Proof:

BASIS: if \(n = 0\) then \(F_0 = 1 - \frac{1}{2^0} = 1 - 1 = 0\)
INDUCTION: Suppose that \(F_n = 1 - \frac{1}{2^n}\) then per construction, \(F_{n+1} = \frac{1}{2} F_n + \frac{1}{2} = \frac{1}{2} - \frac{1}{2}\frac{1}{2^n} + \frac{1}{2} = 1 -\frac{1}{2^{n+1}}\) \(\square\)

The proof technique is called induction¹.

Note that \(F_6 = 0.9843750 \), i.e. we get here a purity coefficient \(0.9843750\) which is almost \(1\) after \(6\) generations only!

During the redaction of blog, I found an interesting article from Lactanet (Réseau Canadien pour l'Excellence Laitière)². Instead of using scientific arguments, the Canadian parliament has decided that an animal is of pure race if its purity coefficient is at least \(87.5\). This corresponds to \(F_3\) only. As a conclusion, I will say that the law is more powerful than the science in Canada.

Key Terms

Crossbreed: A crossbreed is an organism with purebred parents of two different breeds, varieties, or populations.
Offspring: In biology, offspring are the young creation of living organisms, produced either by a single organism or, in the case of sexual reproduction, two organisms.

Wikipedia. Mathematical Induction (last visited Apr. 15, 2022).
Lactanet. Définir la pureté de la race.

Prof. Dr. Jacques E. Boillat is a retired professor for Computer Sciences. Jacques E. Boillat has been studying Mathematics, Theoretical Pysics, and Philosophy at the University of Berne, in Switzerland. He has been teaching at the University of Applied Sciences in Berne and at the University of the Gambia from 2006 to 2020. His wife Lenna Correa Boillat is owner of the Shepherd's Farm in Bato Kunku.

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