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Chapter 3: Problem 35

Assume that Migaloo's albinism is caused by a rare recessive gene. (a) In a mating of two heterozygous, normally pigmented whales, what is the probability that the first three offspring will all have normal pigmentation? (b) What is the probability that the first female offspring is normally pigmented? (c) What is the probability that the first offspring is a normally pigmented female?

Short Answer

Expert verified
a) Three normally pigmented offspring in a row. b) The first female offspring being normally pigmented. c) The first offspring being a normally pigmented female. Answer: a) The probability of three normally pigmented offspring in a row is 27/64. b) The probability of the first female offspring being normally pigmented is 3/8. c) The probability of the first offspring being a normally pigmented female is 3/8.

Step by step solution

01

(a) Probability of three normally pigmented offspring

To calculate this probability, we need to first understand the probability of obtaining a single normally pigmented offspring. Since both parents are heterozygous (one dominant gene and one recessive gene), we can apply the rules of the Punnett square: Heterozygous Parent 1 (Aa) x Heterozygous Parent 2 (Aa): AA Aa Aa aa There is a \(\frac{3}{4}\) chance for each offspring to be normally pigmented (having either AA or Aa genotype) and a \(\frac{1}{4}\) chance to have albinism (having the aa genotype). Now, we can calculate the probability of obtaining three normally pigmented offspring in a row: P(normally pigmented offspring) = \(\left(\frac{3}{4}\right)^3 = \frac{27}{64}\) So, the probability of the first three offspring all having normal pigmentation is \(\frac{27}{64}\).
02

(b) Probability of the first female offspring being normally pigmented

To find this probability, we will first look at the probability of a female offspring. Since the sex of the offspring is determined by the X or Y chromosomes (XX for female and XY for male), and the male contributes either X or Y, we find that the likelihood of having a female offspring is \(\frac{1}{2}\). Now, let's combine this with the probability of obtaining a normally pigmented offspring, which is \(\frac{3}{4}\). P(normally pigmented female offspring) = \(\frac{1}{2} \cdot \frac{3}{4} = \frac{3}{8}\) So, the probability of the first female offspring being normally pigmented is \(\frac{3}{8}\).
03

(c) Probability of the first offspring being a normally pigmented female

This is a combination of the probabilities calculated in parts (a) and (b). However, since we are looking specifically for first offspring with both traits (female and normally pigmented), we can use the probability calculated in part (b). P(normally pigmented female offspring) = \(\frac{3}{8}\) Thus, the probability that the first offspring is a normally pigmented female is \(\frac{3}{8}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Punnett Square
The Punnett square is a visual tool used in genetics to predict the genotypes of offspring from a particular cross. By arranging the alleles from each parent, we can show all the possible combinations for their offspring's genotypes.

The square is comprised of a grid where the alleles from one parent are placed on the top row and those from the second parent are placed on the side column. The intersection of these rows and columns indicates the possible genotypes for the offspring.

For example, when two heterozygous whales (Aa) mate, each can provide allele A or allele a. The Punnett square would look like this:
  • Top: A a
  • Side: A a

If we fill out the squares, we end up with the following genotypic combinations:
  • AA (homozygous dominant)
  • Aa (heterozygous)
  • Aa (heterozygous)
  • aa (homozygous recessive)

This simple grid helps students understand how genotypes can be inherited and the likelihood of each, thereby providing the basic knowledge needed to answer genetics probability questions.
Homozygous and Heterozygous Genotypes Explained
Genotypes represent the genetic make-up of an organism, particularly regarding a specific characteristic. A homozygous genotype has two identical alleles for a particular gene, either both dominant (AA) or both recessive (aa). In contrast, a heterozygous genotype (Aa) has one dominant and one recessive allele.

If an allele is dominant, it can mask the expression of the other, which means that the traits associated with the dominant allele will appear in the organism, even if only one dominant allele is present. However, if an organism is homozygous recessive (aa), the recessive trait will only appear when no dominant allele is present.

In the whale example, AA and Aa genotypes will both result in normally pigmented whales because pigment is the dominant trait. Only the aa genotype gives the recessive trait of albinism. Understanding these concepts helps students solve probability questions involving genetic crosses.
Recessive Genes Inheritance Patterns
Recessive genes require two copies for the trait they represent to be expressed. This means an organism must have a homozygous recessive (aa) genotype to show a recessive trait.

The inheritance of recessive genes follows particular patterns. For a child to express a recessive trait like albinism in whales, both parents must carry at least one recessive allele (Aa or aa). If one parent is homozygous dominant (AA), none of the offspring will show the recessive trait, though they can still be carriers (Aa).

Understanding the inheritance patterns of recessive genes not only applies to scenarios like Migaloo's albinism but also to many human genetic conditions. Knowing how these genes are passed down can inform students on the probability of inheriting certain disorders or traits.

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Most popular questions from this chapter

In this chapter, we focused on the Mendelian postulates, probability, and pedigree analysis. We also considered some of the methods and reasoning by which these ideas, concepts, and techniques were developed. On the basis of these discussions, what answers would you propose to the following questions: (a) How was Mendel able to derive postulates concerning the behavior of "unit factors" during gamete formation, when he could not directly observe them? (b) How do we know whether an organism expressing a dominant trait is homozygous or heterozygous? (c) In analyzing genetic data, how do we know whether deviation from the expected ratio is due to chance rather than to another, independent factor? (d) since experimental crosses are not performed in humans, how do we know how traits are inherited?

In assessing data that fell into two phenotypic classes, a geneticist observed values of \(250: 150 .\) She decided to perform a \(\chi^{2}\) analysis by using the following two different null hypotheses: (a) the data fit a 3: 1 ratio, and (b) the data fit a 1: 1 ratio. Calculate the \(\chi^{2}\) values for each hypothesis. What can be concluded about each hypothesis?

Albinism, lack of pigmentation in humans, results from an autosomal recessive gene (a). Two parents with normal pigmentation have an albino child. (a) What is the probability that their next child will be albino? (b) What is the probability that their next child will be an albino girl? (c) What is the probability that their next three children will be albino?

Two true-breeding pea plants were crossed. One parent is round, terminal, violet, constricted, while the other expresses the respective contrasting phenotypes of wrinkled, axial, white, full. The four pairs of contrasting traits are controlled by four genes, each located on a separate chromosome. In the \(\mathrm{F}_{1}\) only round, axial, violet, and full were expressed. In the \(\mathrm{F}_{2},\) all possible combinations of these traits were expressed in ratios consistent with Mendelian inheritance. (a) What conclusion about the inheritance of the traits can be drawn based on the \(\mathrm{F}_{1}\) results? (b) In the \(\mathrm{F}_{2}\) results, which phenotype appeared most frequently? Write a mathematical expression that predicts the probability of occurrence of this phenotype. (c) Which \(\mathrm{F}_{2}\) phenotype is expected to occur least frequently? Write a mathematical expression that predicts this probability. (d) In the \(F_{2}\) generation, how often is either of the \(P_{1}\) phenotypes likely to occur? (e) If the \(F_{1}\) plants were testcrossed, how many different phenotypes would be produced? How does this number compare with the number of different phenotypes in the \(\mathrm{F}_{2}\) generation just discussed?

In a study of black guinea pigs and white guinea pigs, 100 black animals were crossed with 100 white animals, and each cross was carried to an \(\mathrm{F}_{2}\) generation. In 94 of the crosses, all the \(\mathrm{F}_{1}\) offspring were black and an \(\mathrm{F}_{2}\) ratio of 3 black: 1 white was obtained. In the other 6 cases, half of the \(\mathrm{F}_{1}\) animals were black and the other half were white. Why? Predict the results of crossing the black and white \(\mathrm{F}_{1}\) guinea pigs from the 6 exceptional cases.
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