Problem 39 - Algebra and Trigonometry

Question

Genetics Many genetic traits are controlled by two genes, one dominant and one recessive. In Gregor Mendel's original experiments with peas, the genes controlling the height of the plant are denoted by T (tall) and t (short). The gene T is dominant, so a plant with the genotype (genetic makeup) TT or Tt is tall, whereas one with genotype tt is short. By a statistical analysis of the offspring in his experiments, Mendel concluded that offspring inherit one gene from each parent and that each possible combination of the two genes is equally likely. If each parent has the genotype Tt, then the following chart gives the possible genotypes of the offspring: Find the probability that a given offspring of these parents will be (a) tall or (b) short.

Step-by-Step Solution

Answer

(a) 75% tall; (b) 25% short.

Step 1: Understand Possible Genotypes

Each parent has the genotype Tt. Each offspring inherits one gene from each parent. Therefore, the possible genotype combinations for an offspring are: TT, Tt, Tt, and tt. There are four potential combinations in total.

Step 2: Determine Tall Offspring

To determine the probability that an offspring is tall, we need to consider all genotypes that result in a tall plant. The possible tall genotypes are TT and Tt. Looking at the combinations from Step 1, there are three outcomes (TT, Tt, and Tt) that result in a tall plant.

Step 3: Calculate Probability for Tall Offspring

There are 3 outcomes that produce a tall offspring out of 4 total possible outcomes. Therefore, the probability that an offspring is tall is \( \frac{3}{4} \) or 75%.

Step 4: Determine Short Offspring

For the offspring to be short, the genotype must be tt. Out of the four total possible genotypes for the offspring, only one genotype, tt, results in a short plant.

Step 5: Calculate Probability for Short Offspring

There is 1 outcome that results in a short offspring out of 4 possible outcomes. Therefore, the probability that an offspring is short is \( \frac{1}{4} \) or 25%.

Key Concepts

Genetics

Genetics is the study of how traits are passed from parents to offspring through genes. These traits can range from eye color to disease susceptibility. Each individual inherits two versions of each gene, known as alleles, one from each parent. These alleles can be identical or different, influencing the visible traits, also known as the phenotype, that an individual displays.

In the realm of genetics, it's important to differentiate between dominant and recessive alleles. Dominant alleles, when present, tend to mask the effect of recessive alleles. This means that the presence of a dominant allele will determine the phenotype, even if only one copy is present.

Understanding how these alleles work together to produce observable traits is key to studying genetic inheritance. By predicting allele combinations, geneticists can estimate the likelihood of certain traits appearing in the next generation. This forms the foundation of predictive genetic analysis.

Gregor Mendel

Gregor Mendel, often referred to as the "father of modern genetics," was a 19th-century scientist and Augustinian monk. His pioneering work on pea plants laid the foundation for the scientific study of heredity. Mendel chose pea plants for his experiments due to their ease of breeding and clearly observable contrasting traits, such as plant height and seed color.

Mendel's experiments were groundbreaking. He cross-pollinated pea plants with distinct traits and meticulously recorded how these traits manifested in successive generations. Through his observations, Mendel formulated several key principles of inheritance:

• Law of Segregation: During reproduction, alleles separate so each offspring inherits one allele from each parent.
• Law of Independent Assortment: Traits are passed independently of one another, leading to a variety of genetic combinations.

Mendel's laws have become central to our understanding of genetic inheritance and explain why traits may skip generations or appear unexpectedly.

Genotype

A "genotype" refers to the genetic makeup of an individual, represented by the pair of alleles present for a specific gene. It is important to note that genotypes are not visible traits themselves but rather the genetic blueprint that determines them.

Genotypes are often expressed with letters. For example, in Mendel's pea plants experiment, "T" might represent the dominant allele for tallness, while "t" represents the recessive allele for shortness. Thus, the possible genotypes for plant height could be:

• TT: Homozygous dominant, results in a tall plant.
• Tt: Heterozygous, still results in a tall plant because "T" is dominant.
• tt: Homozygous recessive, results in a short plant.

Understanding the genotype helps in predicting the phenotype, which conveys why an organism looks and behaves the way it does.

Dominant and Recessive Traits

Dominant and recessive traits are a fundamental concept in understanding genetic inheritance. These traits arise from different forms of a gene, known as alleles.

A dominant trait is controlled by a dominant allele, and it only requires one copy of this allele for the trait to be expressed. This means that an organism with either one or two copies of the dominant allele will show the dominant trait. Conversely, a recessive trait requires two copies of a recessive allele for the trait to be exhibited in the organism.

For instance, in the context of Mendel's pea plant experiments:

• T (tall) is the dominant allele. A plant with genotypes TT or Tt will be tall.
• t (short) is the recessive allele. A plant needs to have the genotype tt for it to be short.

The concept of dominant and recessive alleles explains how traits can be "hidden" in one generation and "reappear" in another, a phenomenon Mendel observed in his experiments.