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The color of a persons eyes is determined by a single pair of genes.   If they are both blue-eyed genes, then the person will have blue eyes.   If they are both brown-eyed genes, then the person will have brown eyes.   Also, if one of them is a blue-eyed gene and the other a brown-eyed gene, then the person will have brown eyes.

A newborn child independently receives one gene from each of his parents and the gene it receives from a parent is equally likely to be either of the two eye genes of that parent.

Suppose that Joe and both his parents have brown eyes, but Joe's sister and Joe's wife Claudia have blue eyes.

(a) What is the probability that Joe possesses a blue-eyed gene?

(b) What is the probability that Joe and Claudia’s first child will have blue eyes?

(c) If Joe and Claudia’s first child has brown eyes, what is the probability that their next child will also have brown eyes? (Hint: This is not the same as one minus the answer to part (b)).

How to solve part(c)? Suppose we let E1=(the first child has brown eyes),E2=(the next child has brown eyes) How to calculate Pr(E2|E1)?

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closed as off-topic by TZakrevskiy, Graham Kemp, jameselmore, user147263, graydad Sep 15 '15 at 2:54

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  • $\begingroup$ What did you try to solve this question? Did it work? Where do you get stuck? $\endgroup$ – TZakrevskiy Sep 14 '15 at 23:55
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To get you started:

Let A be the allele for brown eyes and a the allele for blue eyes. If an individual has genotype AA, Aa, or aA --one allele from each parent--then their phenotype is brown eyes. The only genotype with phenotype blue eyes is genotype aa.

Now :

  • What is the genotype of Joe's wife? aa, as that is the only genotype for phenotype blue eyes

  • Both parents have phenotype brown eyes and thus their genotypes are either AA, Aa or aA. But as Joe's sister has genotype aa, what must be the genotype of both parents? Both of Joe's parents must have genotype Aa or aA.

  • Thus with equal probability, Joe's genotype is AA, Aa, aA. In other words, with probability $2/6 = 1/3$ Joe has passes on an a allele, the allele for blue eyes

Can you take it from here?

I strongly suggest drawing a genetic diagram, showing the genotype for each individual.

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  • $\begingroup$ Thank you. Following your tips, I try to solve (b) in this way: $\endgroup$ – Emily Sep 15 '15 at 0:14
  • $\begingroup$ Thank you. Following your tips, I try to solve (b) in this way: If Joe has the genotype of Aa or aA, and gives the a allele to his child, with Claudia's genotype being aa, then their first child can have blue eyes. Since the probability of Joe's genotype being Aa or aA is 2/3 and the probability of giving the a allele from the genotype Aa or aA is 1/2. The probability of Joe and Claudia's first child to have blue eyes is 2/3*1/2=1/3. Does it work? $\endgroup$ – Emily Sep 15 '15 at 0:29
  • $\begingroup$ Yes, that's fine. $\endgroup$ – Simon S Sep 15 '15 at 1:46
  • $\begingroup$ I notice that you mentioned in your answer that the probability of Joe having an a allele is 2/6=1/3, but I think it is 2/3. $\endgroup$ – Emily Sep 15 '15 at 2:08
  • $\begingroup$ I notice that you mentioned in your answer that the probability of Joe having an a allele is 2/6=1/3, but I think it is 2/3, because among the three possible genotype for Joe (AA,Aa,aA), when Joe's genotype is Aa or aA, Joe has an a allele, that is 2 out of 3. $\endgroup$ – Emily Sep 15 '15 at 2:15

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