The seeds in bush bean pods are each the product of an independent fertilization event. Green seed color is dominant to white seed color in bush beans. If a heterozygous plant with green seeds self-fertilizes, what is the probability that 6 seeds in a single pod of the progeny plant will consist of all green seeds?

Verified step by step guidance

1. Understand the genetic basis of the problem: Green seed color (G) is dominant to white seed color (g). A heterozygous plant (Gg) has one dominant allele and one recessive allele. When self-fertilized, the offspring will follow Mendel's laws of inheritance.
2. Determine the genotypic ratio from a heterozygous (Gg) self-fertilization: Use a Punnett square to calculate the offspring genotypes. The possible combinations are GG, Gg, Gg, and gg, resulting in a genotypic ratio of 1:2:1 (1 GG, 2 Gg, 1 gg).
3. Translate the genotypic ratio into phenotypic probabilities: Both GG and Gg result in green seeds, while gg results in white seeds. Therefore, the probability of a green seed is 3/4, and the probability of a white seed is 1/4.
4. Apply the binomial probability formula to calculate the probability of all 6 seeds being green: The formula is \( P = \binom{n}{k} p^k (1-p)^{n-k} \), where \( n \) is the total number of trials (6 seeds), \( k \) is the number of successes (6 green seeds), \( p \) is the probability of success (3/4), and \( 1-p \) is the probability of failure (1/4).
5. Substitute the values into the formula: \( P = \binom{6}{6} (3/4)^6 (1/4)^0 \). Simplify the expression to calculate the final probability of all 6 seeds being green.

Key Concepts

Mendelian Genetics

Mendelian genetics is the study of how traits are inherited through generations based on the principles established by Gregor Mendel. It involves understanding dominant and recessive alleles, where dominant traits mask the expression of recessive traits. In this case, green seed color is dominant over white, meaning that a plant with at least one green allele will exhibit green seeds.

Punnett Square

A Punnett square is a diagram used to predict the genetic makeup of offspring from a particular cross. It helps visualize the possible combinations of alleles from the parents. For a heterozygous plant (Gg), the Punnett square would show the probabilities of producing offspring with green (GG or Gg) or white (gg) seeds, which is essential for calculating the likelihood of all seeds being green.

Binomial Probability

Binomial probability refers to the likelihood of a specific number of successes in a fixed number of independent trials, each with the same probability of success. In this scenario, the probability of obtaining green seeds can be calculated using the binomial formula, where the success is defined as producing a green seed. This concept is crucial for determining the probability that all 6 seeds in a pod will be green.