Problem 60 - Calculus for Biology and Medicine

Question

Assume a \(1: 1\) sex ratio. A woman who is a carrier of hemophilia has two daughters and two sons with a man who is not hemophilic. What is the probability that one daughter is not a carrier, one daughter is a carrier, one son is hemophilic, and one son is not hemophilic?

Step-by-Step Solution

Answer

The probability is 0.25.

Step 1: Determine the Genotypes

A woman who is a carrier of hemophilia can be represented by the genotype \(X^hX\), where \(X^h\) is the affected X chromosome. The man who is not hemophilic will have the genotype \(XY\). Hence, we denote the possible combinations of children genotypes: for daughters, \(X^hX\) or \(XX\); for sons, \(X^hY\) or \(XY\).

Step 2: Determine Probabilities for Daughters

Each daughter has a 50% chance of inheriting either the \(X^h\) or \(X\) chromosome from the mother. Hence, one daughter's probability of being a carrier (\(X^hX\)) is 0.5, and not being a carrier (\(XX\)) is also 0.5.

Step 3: Determine Probabilities for Sons

Each son has a 50% chance of inheriting the \(X^h\) (hemophilic) or \(X\) (non-hemophilic) chromosome from the mother. Therefore, one son's probability of being hemophilic (\(X^hY\)) is 0.5, and not being hemophilic (\(XY\)) is also 0.5.

Step 4: Calculate Combined Probability

To find the probability of having one daughter as a carrier, one daughter not as a carrier, one son hemophilic, and one son not hemophilic, calculate: \( 0.5 \times 0.5 \times 0.5 \times 0.5 = 0.0625 \).

Step 5: Consider Different Arrangements

The different possible arrangements are 4: one daughter being a carrier, one not; one son being hemophilic, and one not (e.g., daughter1-carrier, daughter2-not, son1-hemophilic, son2-not). Therefore, multiply the probability of one specific arrangement by the number of ways (4): \(0.0625 \times 4 = 0.25\).

Key Concepts

Sex-linked traits

In genetics, some traits are linked to the genes located on sex chromosomes, which are the X and Y chromosomes. These are referred to as sex-linked traits. The most well-known example of sex-linked traits are those linked to the X chromosome, since the X chromosome carries more genes than the much smaller Y chromosome. Thus, males (XY) and females (XX) tend to exhibit these traits differently due to their different combinations of X and Y chromosomes.
A key sex-linked trait is hemophilia, a disorder that affects blood clotting. Since the gene for hemophilia is located on the X chromosome, it is termed X-linked. In this case, females can be carriers (possessing one affected X chromosome), while males who inherit the affected X chromosome will express hemophilia because they have only one X chromosome.
Understanding sex-linked traits helps us see why certain conditions, like hemophilia, occur more frequently in males. In the exercise, the mother is a carrier, so she has one affected and one normal X chromosome, while the father isn't affected and provides a normal X or Y chromosome to their children.

Probability calculation

Probability calculation is a statistical method used to predict how likely certain genetic outcomes are, given specific starting conditions. In genetics, calculations like these help estimate the likelihood of offspring inheriting particular traits from their parents.
For our exercise, each child has a 50/50 chance of inheriting various combinations of genes for hemophilia from the parents. This breaks down into each daughter having a 50% chance of being a carrier or non-carrier for hemophilia, and each son having a 50% chance of being hemophilic or non-hemophilic.
To find the probability of one daughter being a carrier, one daughter not being a carrier, one son being hemophilic, and one son not being hemophilic, you multiply these individual probabilities: \(0.5 imes 0.5 imes 0.5 imes 0.5 = 0.0625\).
This result represents the probability of this specific genetic outcome occurring.

Punnett square

A Punnett square is a handy diagram used in genetics to determine the probability of an offspring inheriting particular alleles from its parents. It visualizes how different allele combinations can occur between the parental genotypes.
In our scenario, the mother has a genotype of \(X^hX\) as she is a carrier, while the father has a genotype of \(XY\) as he is not affected. By setting up a Punnett square, you can predict the possible genetic outcomes for their children. For instance, for daughters, the combinations result in either \(X^hX\) (carrier) or \(XX\) (non-carrier). For sons, the options are \(X^hY\) (hemophilic) and \(XY\) (non-hemophilic).
Thus, the Punnett square offers a simple yet powerful way to visualize and calculate the likelihood of different genetic trait distributions, which is key in understanding the probability distributions addressed in the exercise.