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 3 green and 3 white seeds?

Verified step by step guidance

1. Step 1: Understand the genetic basis of the problem. Green seed color (G) is dominant to white seed color (g). A heterozygous plant (Gg) self-fertilizes, meaning the offspring will follow Mendelian inheritance for a monohybrid cross.
2. Step 2: Determine the genotype probabilities from the self-fertilization of a heterozygous plant (Gg × Gg). Using a Punnett square, the offspring genotypes will be: GG (25%), Gg (50%), and gg (25%). Since green is dominant, the phenotypic probabilities are: green seeds (75%) and white seeds (25%).
3. Step 3: Recognize that the seeds in the pod are independent events. The probability of each seed being green or white follows the binomial distribution, where the probability of success (green seed) is 0.75 and the probability of failure (white seed) is 0.25.
4. Step 4: Use the binomial probability formula to calculate the probability of exactly 3 green seeds and 3 white seeds in a pod of 6 seeds. The formula is: \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), where \( n \) is the total number of trials (6 seeds), \( k \) is the number of successes (3 green seeds), \( p \) is the probability of success (0.75), and \( 1-p \) is the probability of failure (0.25).
5. Step 5: Substitute the values into the formula: \( P(X = 3) = \binom{6}{3} (0.75)^3 (0.25)^3 \). Calculate the binomial coefficient \( \binom{6}{3} \), which represents the number of ways to choose 3 green seeds out of 6, and then multiply by the probabilities raised to their respective powers.

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 includes concepts such as 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 genotypic and phenotypic outcomes of a genetic cross. It helps visualize the possible combinations of alleles from the parents. For a heterozygous plant (Gg), the Punnett square can show the expected ratios of offspring, which is essential for calculating the probability of obtaining specific seed colors in the progeny.

Binomial Probability

Binomial probability refers to the likelihood of obtaining a fixed number of successes in a fixed number of independent trials, each with two possible outcomes. In this scenario, the probability of getting 3 green and 3 white seeds from 6 total seeds can be calculated using the binomial formula, which accounts for the different combinations of outcomes and their respective probabilities.