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A gene is composed of two alleles, each can be of type $A$ or $a$. In the population there are three types of individuals:
$$AA$$ $$Aa$$

$$aa$$
Each parent transmits to his son one of the two alleles chosen at random. Knowing that initially the proportions are:

$$AA = 1/3$$

$$Aa = 1/5$$

$$Aa = 7/15$$ What are the proportions of type $AA', Aa', aa'$ in the next generation?

MY TRY

$$P(\phi A) = P(\phi A | AA)\cdot P(AA)+P(\phi A | Aa)\cdot P(Aa)+P(\phi A | aa)\cdot P(aa) = 0.4333...$$ $$P(\phi A') = 1-P(\phi A) = 0.5666...$$ $$P(A,A') = P(\phi A') \cdot P(\phi A') \approx \color{red}{0.3211}$$ $$P(A,a') = 2 \cdot P(\phi A') \cdot P(\phi A) \approx \color{red}{0.491}$$ $$P(a,a') = P(\phi A) \cdot P(\phi A) \approx \color{red}{0.1878}$$

However, the results should be reversed, like that:

$$P(A,A') = P(\phi A') \cdot P(\phi A') \approx \color{green}{0.1878}$$ $$P(A,a') = 2 \cdot P(\phi A') \cdot P(\phi A) \approx \color{green}{0.491}$$ $$P(a,a') = P(\phi A) \cdot P(\phi A) \approx \color{green}{0.3211}$$

Can you help me figure out where i'm wrong, or to illustrate another method to get the result? Thank you

1 Answers 1

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Not sure what $\phi$ represents in your post.

Let's compute the probability of $AA$ in the next generation:

We note that $AA$ can arise from a marriage of the types $(AA,AA)$, $(AA,Aa)$, $(Aa,AA)$, or $(Aa,Aa)$. Let's just handle each case separately.

Case I $(AA,AA)$. Such a marriage occurs $\frac 19$ of the time and it always results in $AA$ offspring. Thus $\boxed {\frac 19}$

Case II $(AA,Aa)$ such a marriage occurs $\frac 1{15}$ of the time and it results in $AA$ offspring half the time. Thus $\boxed {\frac 1{30}}$

Case III $(Aa,AA)$. By symmetry, $\boxed {\frac 1{30}}$

Case IV $(Aa,Aa)$. Such a marriage occurs $\frac 1{25}$ of the time and results in $AA$ offspring a quarter of the time. Thus $\boxed {\frac 1{100}}$

Combining we see that the probability of $AA$ in the next generation is $$\frac 19+ \frac 1{30}+\frac 1{30}+\frac 1{100}=\frac {169}{900}=\boxed {.18\overline {77}}$$

The other cases are handled similarly.