Question

An alternative to using the expanded binomial equation and Pascal's triangle in determining probabilities of phenotypes in a subsequent generation when the parents' genotypes are known is to use the following equation: \(\frac{n !}{s ! t !} a^{s} b^{t}\) where \(n\) is the total number of offspring, \(s\) is the number of offspring in one phenotypic category, \(t\) is the number of offspring in the other phenotypic category, \(a\) is the probability of occurrence of the first phenotype, and \(b\) is the probability of the second phenotype. Using this equation, determine the probability of a family of 5 offspring having exactly 2 children afflicted with sickle-cell anemia (an autosomal recessive disease \()\) when both parents are heterozygous for the sickle-cell allele.

Short answer

Answer: The probability is \(\frac{135}{512}\).

Step-by-step solution

Calculate the probabilities of each phenotype

To find the probability of children being afflicted or not afflicted with sickle-cell anemia, we need to consider the possible genotypes of the parents. Both parents are heterozygous, which means they carry one normal allele (A) and one sickle-cell allele (S). Therefore, their genotypes are AS. When two heterozygous parents (AS) reproduce, the possible outcomes for their offspring are: 1. AA (normal): Probability = 1/4 2. AS (carrier): Probability = 1/2 3. SA (carrier - same as AS): Probability = 1/4 4. SS (sickle-cell anemia): Probability = 1/4 We will consider the SS genotype as the affected offspring, and we'll group AA, AS, and SA genotypes as non-affected offspring. So, the probabilities are: - Probability of having sickle-cell anemia (a): 1/4 - Probability of not having sickle-cell anemia (b): 3/4

Apply the given equation to determine the probability of having 2 children with sickle-cell anemia

Now, we'll use the equation provided: \(\frac{n !}{s ! t !} a^{s} b^{t}\) Where: - n = total number of offspring (5) - s = number of offspring with sickle-cell anemia (2) - t = number of offspring not having sickle-cell anemia (3) - a = probability of having sickle-cell anemia (1/4) - b = probability of not having sickle-cell anemia (3/4) Plugging in the values in the equation, we have: \(\frac{5 !}{2 ! 3 !} (1/4)^{2} (3/4)^{3}\)

Solve the equation

Let's evaluate the equation step by step: 1. Find the factorials: 5! = 120, 2! = 2, 3! = 6 2. Substitute the factorials into the equation: \(\frac{120}{2 * 6} (1/4)^{2} (3/4)^{3}\) 3. Simplify the factorial fraction: \(\frac{120}{12} (1/4)^{2} (3/4)^{3}\) = 10(1/4)^{2}(3/4)^{3}$ 4. Calculate the probabilities raised to their respective powers: \((1/16)(27/64)\) 5. Multiply the fractions together: \(10(27/1024)\) 6. Simplify the final fraction: \(10(27/1024) = \frac{270}{1024} = \frac{135}{512}\) The probability of a family of 5 having exactly 2 children with sickle-cell anemia when both parents are heterozygous is \(\frac{135}{512}\).

Key concepts

Phenotype Probability

Phenotype probability is an essential concept within genetics, referring to the likelihood of a certain phenotype appearing in the offspring, based on the genetic makeup of the parents. Phenotypes are the observable traits or characteristics, like having sickle-cell anemia.

When calculating the probability of phenotypes, consider the genetic combinations parents can pass onto their offspring. In the case of sickle-cell anemia, which is autosomal recessive, a child needs to inherit two copies of the sickle-cell allele to express the phenotype. If both parents are heterozygous, like in our exercise, they carry one normal allele and one sickle-cell allele, giving rise to different genotypic combinations. Understanding these combinations helps in determining the probability of conditions like sickle-cell anemia. Remember, the practical application of phenotype probability is crucial in genetic counseling and predicting genetic disorders.

Binomial Equation

The binomial equation is a powerful mathematical tool used to calculate the probabilities of different outcomes in genetics, particularly when dealing with traits passed on to multiple offspring. It provides a structured approach to determining how likely certain phenotypes will appear over a set number of offspring.

The general format is: - \[ \frac{n!}{s!t!}a^s b^t \]- \(n\) is the total number of offspring.- \(s\) is the number of offspring with a specific phenotype.- \(t\) is the number with another phenotype.- \(a\) and \(b\) represent the probabilities of each phenotype, respectively.
In our example, we calculated probabilities for sickle-cell anemia. Using factorials and powers of probabilities, the equation determines how likely it is for a certain number of children to inherit specific genetic traits. The equation is helpful because it can handle complex inheritance scenarios with ease, reducing intricate probability calculations to straightforward arithmetic steps.

Mendelian Inheritance

Mendelian inheritance is the foundational theory of genetics that was first described by Gregor Mendel in the 19th century. This theory explains how traits are passed from parents to offspring through discrete units known as genes.

Using simple pea plant experiments, Mendel discovered how traits followed specific patterns of segregation and assortment. In the context of our exercise, Mendelian inheritance principles show us how sickle-cell anemia can be transmitted as an autosomal recessive trait, only appearing when offspring receive two recessive alleles (from both heterozygous parents).

The core rules of Mendelian inheritance include: - **Dominance:** Some alleles are dominant while others are recessive. - **Segregation:** Each parent contributes one of two possible alleles to their offspring. - **Independent Assortment:** Genes for different traits can segregate independently during gamete formation.
These principles provide the framework to predict the genetic makeup of offspring and understand how conditions like sickle-cell anemia can appear within families, key knowledge for genetics students.