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In genetics, traits can be categorized as either dominant or recessive. Dominant traits only require one copy of an allele to be expressed. This means that if an organism has at least one dominant allele in its genotype, the dominant trait will be visible in its phenotype. For instance, if black coat color in cats is dominant, then any cat possessing the allele for black will showcase a black coat.

On the flip side, recessive traits are a bit more elusive. Both alleles must be recessive for the trait to be expressed. This means that the phenotype displaying the recessive trait can only occur if an organism's genotype has two recessive alleles. Therefore, when considering Sam the white cat's offspring, for a kitten to be white, it would need two recessive alleles for coat color. In summary:

• Dominant traits show up with just one allele.
• Recessive traits require two copies to be expressed.

Genotype refers to the genetic makeup of an organism; it's the blueprint in its DNA. In other words, it's the set of genes an organism carries. In Sam the cat's case, Sam's genotype involves alleles that determine coat color – specifically, one allele for a recessive trait (white) and one for a dominant trait (black).

Phenotype, on the other hand, is the physical expression of that genetic code. It includes visible traits like color, shape, and size. So, Sam's phenotype might appear white, but her genotype reveals there’s more to the story when we discover the distribution of her offspring’s coat colors.

To simplify, we can think of genotype as the genetic instructions and phenotype as the executed plan:

• Genotype = genetic makeup (WW, Ww, or ww).
• Phenotype = observable traits (white or black coat).

Mendelian inheritance is a set of principles relating to the transmission of hereditary characteristics from parent organisms to their offspring. Named after Gregor Mendel, who discovered these patterns, it explains how traits are passed down through generations via dominant and recessive alleles.

In the case of Joe's cats, Mendelian inheritance helps us understand the results when crossing a white cat with a black cat. According to Mendelian principles, if a black coat is a dominant trait, we would expect the presence of at least one black allele would make the kitten appear black.

The results from Joe's breeding experiment reflect Mendel's findings through dominance and segregation of alleles. Each kitten gets a set of alleles from each parent, which determines their coat color:

• If they inherit a dominant allele, they will be black.
• They become white only if both inherited alleles are recessive.

This pattern aligns with Mendel's principle of segregation, which states that allele pairs separate during gamete formation, allowing for varied genetic combinations in offspring.