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Genetic inheritance follows specific patterns, each delineating how certain traits are transmitted from parents to offspring through genes. When investigating characteristics such as horse gaits, two basic patterns are widely recognized: complete dominance and incomplete dominance.

In the context of horse gaits, the exercise suggests that trotting is a trait governed by complete dominance. This means when a horse inherits differing alleles for gait (one for trotting and one for pacing), the trait that becomes physically manifest in the horse (the phenotype) is the one governed by what's known as the dominant allele; in this case, trotting. Hence, if a horse possesses at least one allele for the trotting gait, it will exhibit that gait regardless of the second allele.

Complete dominance simplifies genetics to a degree since it implies a clear-cut expression of one trait over another when mixed alleles are present. However, genotypes still play a crucial role in understanding the probability of these traits appearing in the next generation. Incomplete dominance or other inheritance patterns, such as codominance, where both alleles contribute to the phenotype, are not suggested by the exercise and therefore do not apply to this scenario.

Within the genetic framework of dominance, two types of alleles exist: dominant and recessive. A dominant allele, like the one for trotting in horses, is denoted using a capital letter (in this case, 'T'). This allele will mask the presence of a recessive allele when both are present in an organism's genotype, which is the genetic makeup.

Recessive genes, on the other hand, like the one for pacing in horses, are represented with a lowercase letter ('t'). For the recessive trait to be expressed, an organism must inherit two copies of the recessive allele (tt), one from each parent. If an organism inherits one dominant allele and one recessive allele (Tt), the dominant trait will be expressed, whereas the recessive trait will only be expressed if both alleles are recessive.

When analyzing genetic outcomes, it's important to note that an organism with two identical alleles for a trait (such as TT or tt) is termed homozygous, while an organism with two different alleles (Tt) is referred to as heterozygous. The exercise shows a typical scenario where a dominant trait overshadows a recessive one, showcasing the foundational principles of Mendelian genetics.

The genotypic determination of an organism involves the specific combination of alleles that it inherits from its parents. In our horse gait example, the genotype determines whether a horse will trot ('T', dominant) or pace ('t', recessive).

The stallion in the exercise exhibits pacing, a recessive trait, indicating that his genotype must be homozygous recessive (tt) because pacing only manifests if no dominant trots alleles are present. The mares, which trot and can only pass down the dominant gene, have the homozygous dominant genotype (TT). When these two genotypes are crossed (TT x tt), all progeny will have the heterozygous genotype (Tt).

In other words, offspring receive one allele from each parent, and the combination of these alleles (their genotype) ultimately determines their traits (their phenotype). Through genetic crosses, scientists and breeders can predict the outcome of specific trait expressions. Thus, the exercise reinforces understanding the principles that underpin genotypic determination: namely, that an individual's genotype is the blueprint that directs the expression of physical, observable traits.