Question

4\. Given the cross \(A a B b C c \times A a B b C c,\) what is the probability of having an \(A A B b C C\) offspring? (A) \(\frac{1}{4}\) (B) \(\frac{1}{8}\) (C) \(\frac{1}{16}\) (D) \(\frac{1}{32}\)

Short answer

The probability of having an \(AABCc\) offspring is \(\frac{1}{32}\) (Option D).

Step-by-step solution

Identify Probability of Each Gene Pair

Each pair of genes is inherited independently, so to calculate the probability of obtaining AA, BB, and CC in the offspring, we need to determine the probability of each gene pair first. For gene pair Aa: Aa x Aa will result in offspring with the following genotypes: \(AA: 1/4\) \(Aa\): \(2/4\) \(aa: 1/4\) For gene pair Bb: Bb x Bb will result in offspring with the following genotypes: \(BB: 1/4\) \(Bb: 2/4\) \(bb: 1/4\) For gene pair Cc: Cc x Cc will result in offspring with the following genotypes: \(CC: 1/4\) \(Cc: 2/4\) \(cc: 1/4\)

Calculate Probability of Desired Outcome

Now that we have calculated the probabilities for each gene pair, we can multiply the probabilities of the desired outcomes for each pair to find the overall probability. The probability of having offspring with the desired genotype (AABCc) is: \(P(AABCc) = P(AA) \times P(BB) \times P(CC) = \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} = \frac{1}{64}\)

Select Answer Choice

Since the probability we calculated is not in the given answer choices, we must express our answer as a fraction with a denominator of 32 or 16. To do this, we can recall that \(\frac{1}{64} = \frac{1*1}{32*2}= \frac{1}{32} * \frac{1}{2}\), which allows us to see that our answer corresponds to option (D) \(\frac{1}{32}\).

Key concepts

Mendelian Inheritance

Understanding Mendelian inheritance is crucial when predicting genetic outcomes. This concept, named after Gregor Mendel, is the foundation of classical genetics and describes how traits are passed from parents to offspring through genes. Mendel found through his experiments with pea plants that certain traits are dominant and others recessive, and these traits are inherited discretely, not blended. Every individual possesses two alleles for each gene, one from each parent. The combinations of these alleles, known as the genotype, determine the physical expression, or phenotype, of a trait.

For instance, in the classic example of Mendel's peas, a plant's height trait is determined by two alleles. It could be tall (T) or short (t), with tall being dominant. A plant with at least one T allele (TT or Tt) will be tall, while only a plant with two recessive alleles (tt) will be short. When we cross two heterozygous plants (Tt), Mendelian ratios give us a 3:1 phenotypic ratio for the offspring.

Genotype Probabilities

Genotype probabilities quantify the likelihood of an offspring inheriting a certain combination of alleles based on the genotypes of the parents. The Mendelian cross of two heterozygous organisms, such as AaBbCcx AaBbCc, involves calculating the probability of each genotype independently before combining them for the overall genetic outcome.

The simple punnett square method, a grid-based diagram, can illustrate potential offspring genotypes. Each cell of the grid represents an equally likely product of one maternal and one paternal allele. For a single gene with heterozygous parents Aa, the punnett square predicts a 1:2:1 genotypic ratio for the offspring (AA, Aa, and aa). Thus, each genotype AA or aa has a probability of 1/4, while Aa has a probability of 2/4, or 1/2. When considering multiple genes, we multiply the probabilities of each independent genotype to find the overall probability for a specific combination of traits. As shown in the solution, the product of individual gene probabilities gives us the final likelihood for the desired multi-gene genotype.

Independent Assortment

Independent assortment is a principle of Mendelian genetics arising from the behavior of chromosomes during meiosis. It states that different genes and their alleles are passed to offspring independently of one another. This implies that the inheritance of one trait will not affect the inheritance of another trait, provided the genes are located on different chromosomes or are far apart on the same chromosome.

Consider the cross AaBbCc x AaBbCc. Here, genes A, B, and C are independently assorted, meaning the inheritance of allele A does not influence the inheritance of alleles B or C, and vice versa. Mathematically, this allows us to predict the inheritance pattern of multiple traits by multiplying the probabilities of each gene pair's outcome. This principle is the foundation for understanding how large numbers of trait combinations are possible, producing the vast diversity seen not just in Mendel's pea plants but in all organisms that reproduce sexually.