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A certain organism possesses a pair of each of 5 different genes (which we will designate by the first 5 letters of the English alphabet). Each gene appears in 2 forms (which we designate by lowercase and capital letters). The capital letter will be assumed to be the dominant gene in the sense that if an organism possesses the gene pair xX, then it will outwardly have the appearance of the X gene. For instance, if X stands for brown eyes and x for blue eyes, then an individual having either gene pair XX or xX will have brown eyes, whereas one having gene pair xxwill be blue-eyed. The characteristic appearance of an organism is called its phenotype, whereas its genetic constitution is called its genotype. (Thus 2 organisms with respective genotypes aA, bB, cc, dD, ee and AA, BB, cc, DD, ee would have different genotypes but the same phenotype.) In a mating between 2 organisms each one contributes, at random, one of its gene pairs of each type. The 5 contributions of an organism (one of each of the 5 types) are assumed to be independent and are also independent of the contributions of its mate. In a mating between organisms having genotypes aA, bB, cC, dD, eE, and aa, bB, cc, Dd, ee, what is the probability that the progeny will (1) phenotypically, (2) genotypically resemble

(a) the first parent;

(b) the second parent;

(c) either parent;

(d) neither parent?

Now assuming that genes of different letter types do not intermix ( e.g Ab,Ba etc) , we have (4)^5 possibilities for the child.(aa,AA,Aa,aA for the first slot and so on). Now to resemble the phenotype of the first parent the child should bear the following genetic properties

i)aA,Aa,AA for the first slot

ii)Bb,BB,bB for the second slot

iii)cC,CC,Cc for the third slot

iv) dD,Db,DD for the fourth slot

v) eE,Ee,EE for the fifth slot

Now there are 3^5 possible outcomes that result in the child having similar phenotype as the first parent. So the probability of the first outcome would be (0.75)^5. I would like to know if I am missing something in my approach to this problem.

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Yes, you're missing something. This is the probability that a person with genes uniformly randomly selected from all possible genes would resemble the first parent. But you want that probability for the progeny of these two parents; so for each gene you need to calculate the probability that one of the four equally likely combinations of the parents' genes will yield one of the three combinations that you identified, and then form the product over all five genes.

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  • $\begingroup$ How do we calculate the probability that one of the four equally likely combinations of the parents' genes will yield one of the three combinations? Do we use Bayes' Theorem here? For eg P(Aa)=P(Aa|Aa)*P(Aa)+P(Aa|AA)*P(AA)+P(Aa|aA)*P(aA)+P(Aa|aa)*P(aa) and doing the same for aA and AA? $\endgroup$ – adhok Sep 11 '15 at 8:05
  • $\begingroup$ @adhok: No, the parents' genes are given; no need to treat them probabilistically. $\endgroup$ – joriki Sep 11 '15 at 8:06

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