The genes $A$ and $a$ correspond to a genetic trait with complete dominance. Two heterozygous $Aa$ individuals mate. One of their offspring is chosen at random. That offspring, an individual who is of dominant phenotype, mates with another individual heterozygous for the same trait. When taking from the new offspring an individual that is of dominant phenotype, what is the probability that it is also of dominant $AA$ genotype?
gene: codes for a certain trait
allele: variations of a gene (in this case A and a)
homozygous: both alleles are the same (AA or aa)
heterozygous: alleles are different (Aa)
genotype: alleles carried by an individual (AA, Aa, aA, aa)
phenotype: expression of a trait coded for by a gene
dominant phenotype: in heterozygous individuals, the trait expressed by the dominant gene (A) is expressed
dominant genotype: both alleles are for the dominant trait (AA)
Best Answer
We have an offspring of two heterozygous individuals whose phenotype is the dominant one. Since their offspring are equally likely to be AA, Aa, aA or aa (with the alleles listed in order of which parent they are inherited from), this offspring has a $2/3$ chance of being heterozygous.
So, $2/3$ of the time you cross a heterozygous individual with another; again a dominant offspring has a $2/3$ chance of being heterozygous. $1/3$ of the time you cross heterozygous with homozygous dominant, and all the offspring will show the dominant phenotype but half of them will be heterezygous.
Overall, the probability of being heterozygous is $\frac23\times\frac23+\frac13\times\frac12=\frac{11}{18}$. So the probability of being homozygous is $\frac{7}{18}$.
Of course, this will change if you observe another offspring of the second crossing to exhibit the recessive phenotype; this will prove that your first cross was heterozygous and the probability will then be one third.