The following \(F_{2}\) results occur from a typical dihybrid cross: $$\begin{array}{lll} \text { purple: } & A_{-} B_{-} & 9 / 16 \\ \text { white: } & a a B_{-} & 3 / 16 \\ \text { white: } & A_{-} b b & 3 / 16 \\ \text { white: } & a a b b & 1 / 16 \end{array}$$ If a double heterozygote \((A a B b)\) is crossed with a fully recessive organism (aabb), what phenotypic ratio is expected in the offspring?

The Punnett square is a visual tool in genetics used to predict the possible genotypes of offspring from a cross between two parents. In a dihybrid cross, like the one provided in the exercise, we're examining two distinct traits, each represented by a different gene. For a parent with the genotype AaBb, which is heterozygous for both traits, the possible gametes are AB, Ab, aB, and ab.

These combinations are listed on one side of a 4x4 Punnett square, mirroring the unique pairing possibilities these gametes have with those from the other parent. If the other parent is homozygous recessive (aabb), then their only possible gamete is ab, which occupies the opposite side of the Punnett square.

Once the Punnett square is set up, we fill it in by combining each gamete from one parent with each gamete from the other parent. This step-by-step approach visualizes the probabilities of genotypes for each offspring. Even though the math behind the Punnett square is simple probabilities, it greatly enhances our understanding of how traits are passed down.

The phenotypic ratio describes the proportion of offspring with different traits that result from a genetic cross. It's determined by examining the offspring's outward characteristics (phenotypes) rather than their genetic makeup (genotypes). In the provided exercise, we're comparing two phenotypes: purple and white.

After filling out the Punnett square and identifying the genotypes, we translate these into phenotypes based on dominance relationships. In the case of the A and B alleles, the dominant allele masks the presence of the recessive allele, leading to the purple phenotype whenever one or more dominant alleles (A or B) are present.

By calculating how many squares within the Punnett square represent each phenotype, we can establish the phenotypic ratio. It's important to recognize that while the genotypic ratio can be complex in a dihybrid cross due to multiple gene interactions, the phenotypic ratio simplifies this complexity into observable results.

Genetics is the study of heredity and variation in living organisms. It explains how traits are passed from parents to offspring through genes. Genes, which are made up of DNA, act as instructions for making molecules called proteins.

In sexual reproduction, organisms inherit two sets of genes, one from each parent. These genes come in different variants, called alleles, which can be dominant or recessive. The combination of alleles determines an individual's genotype. The genotype, along with environmental factors, then influences the phenotype—the observable traits.

The exercise we've been looking at involves a dihybrid cross, which looks at the inheritance of two different genes. Genetics helps us understand the outcome of such crosses, predicting traits of offspring and allowing us to delve into the probabilities and patterns of inheritance, a fundamental aspect of biology.