Question

In a family of six children, where one grandparent on either side has red hair, what mathematical expression predicts the probability that two of the children have red hair?

Short answer

Answer: The mathematical expression for the probability is 15 × x^2 (1-x)^4, where x represents the probability of a child having red hair and (1-x) represents the probability of a child not having red hair.

Step-by-step solution

Identify the problem type

This exercise is a combination problem, dealing with the probability of having two out of six children with red hair. We will use the binomial coefficient to find the number of different arrangements and compute the probability.

Using binomial coefficient

In a family of six children, where one grandparent on either side has red hair, we will use binomial coefficient to find the different combinations of having two children with red hair. The binomial coefficient for finding two successes (red hair) in six trials (children) is denoted by C(6,2) or 6!/[2!(6-2)!].

Compute the binomial coefficient

Now, we will compute the binomial coefficient C(6, 2): C(6, 2) = 6!/(2!(6-2)!) = 6!/(2!(4)!) = (6 × 5 × 4 × 3 × 2 × 1)/((2 × 1) × (4 × 3 × 2 × 1)) = (6 × 5)/(2) = 15 So, there are 15 different ways to arrange two children with red hair among the six siblings.

Compute the probability expression

Let x represent the probability of a child having red hair. Then, the probability of a child not having red hair is (1-x). Now, we have to compute the probability expression for two of the six children having red hair. For each of these 15 combinations, the probability of two children with red hair and four children without red hair is x^2 (1-x)^4. Since these combinations are mutually exclusive, we should sum the probabilities of each combination: Probability = C(6, 2) × x^2 (1-x)^4 = 15 × x^2 (1-x)^4 This expression represents the probability that two of the six children have red hair in a family where one grandparent on either side has red hair.

Key concepts

Binomial Coefficient

The binomial coefficient, symbolized by the notation C(n, k) or sometimes as choose k, is a fundamental concept in probability and combinatorics, which plays a crucial role in calculating the number of ways 'k' successes can occur in 'n' trials. It's akin to determining how many different combinations of a group can be formed.

For our exercise involving genetics, the binomial coefficient helps us to figure out the various ways two children out of six can inherit a specific trait, like red hair, from their grandparents. The mathematical expression of the binomial coefficient for two successes in six trials is choose 2 which simplifies to 15 combinations after calculation. Each combination represents a unique set of children among the six who could possess the trait in question. Understanding and computing the binomial coefficient is a cornerstone in the study of genetic probability, as it allows us to quantify the patterns in which traits are passed down through generations.

Improving upon this exercise might involve visual aids such as family trees or Punnett squares to enhance the comprehension of how these combinations relate to real-world genetics scenarios.

Inheritance of Traits

In genetics, the inheritance of traits refers to how certain characteristics are transmitted from parents to their offspring through genes. Each parent contributes one allele for every gene, and the combination of these alleles determines the traits of the children. This process is governed by Mendelian inheritance patterns, including dominant and recessive traits.

In the given exercise, having red hair is likely considered a recessive trait, which means a child needs to inherit the red hair allele from both parents to express that phenotype. Our exercise simplifies the situation by not considering the specific odds associated with inheriting the red haired allele; instead, it defines the probability as 'x'. When considering the improvement of students' understanding, it's helpful to explain that genetic traits aren't always a simple matter of dominant or recessive, as some traits are influenced by multiple genes (polygenic) or can exhibit incomplete dominance and co-dominance.

Highlighting real inheritance patterns and how they can deviate from idealized models can provide students with a more nuanced appreciation of genetic variation.

Genetic Probability Calculations

The calculations of genetic probabilities involve determining the likelihood of certain traits appearing in offspring, based on the genetic makeup of their parents. In our exercise, the focus is on finding the probability of having a certain number of children with a trait, given the odds of any one child having that trait.

To do this, we combine our understanding of the binomial coefficient with the laws of probability. We calculate the probability of two children having red hair and four not having it, then multiply the probability of that specific outcome by the number of ways it can occur, which we found to be 15. The resulting expression: \(15 \times x^2(1-x)^4\) builds the foundation for predicting complex genetic outcomes.

For a more intuitive understanding, reiterating the concepts with concrete examples or interactive simulations can significantly help. Engaging with scenarios where probabilities change based on different parental genotypes, or where students can adjust 'x' to see how it affects the outcome, reinforces the relationship between inheritance and probability in a tangible way.