Question

Suppose that a disease is inherited via a sex-linked mode of inheritance so that a male offspring has a \(50 \%\) chance of inheriting the disease, but a female offspring has no chance of inheriting the disease. Further suppose that \(51.3 \%\) of births are male. What is the probability that a randomly chosen child will be affected by the disease?

Short answer

The probability that a randomly chosen child will be affected by the disease is 25.65% (or 0.2565).

Step-by-step solution

Understanding the problem

The problem describes a scenario where a disease is inherited in a sex-linked manner. The inheritance chance for male offspring is given as 50%, while female offspring cannot inherit the disease. The probability of a male birth is 51.3%. We need to calculate the overall probability of a randomly chosen child having the disease.

Calculating the probability for male offspring

Since only male offspring can inherit the disease and the chance of a male inheriting the disease is 50%, we multiply the probability of a child being male by the probability of a male inheriting the disease. Probability of a male child being affected is the product of the probability of being male (51.3% or 0.513) and the probability of a male inheriting the disease (50% or 0.5).

Calculating the overall probability of a child having the disease

To find the total probability that a randomly chosen child will have the disease, we add the probability of a male child having the disease to the probability of a female child having the disease (which is 0, because females cannot inherit the disease in this scenario). Thus, the overall probability is simply the probability calculated in Step 2.

Key concepts

Sex-Linked Inheritance

Sex-linked inheritance is a mode of genetic transmission where certain traits or diseases are carried on sex chromosomes, specifically the X and Y chromosomes. This is in contrast to autosomal inheritance, where traits are carried on non-sex chromosomes. What makes sex-linked inheritance particularly important in genetics is that males and females can be affected differently by the same genetic conditions.

Men have one X and one Y chromosome (XY), while women have two X chromosomes (XX). If a disease is linked to the X chromosome, it often has a more pronounced effect in males since they have only one X chromosome, and no second X to potentially offset the genetic mutation with a healthy allele. Females, however, with their two X chromosomes, tend to be carriers more often than they are affected, as one healthy allele on one X chromosome can often compensate for a diseased allele on the other.

A classic example of this type of inheritance is hemophilia, where a single mutated gene on the X chromosome can lead to the disease in males. Females might carry the gene, and only express the disease in the rare case where they have the mutated gene on both of their X chromosomes. The exercise provided is describing such a disease by stating that male offspring have a 50% chance of inheriting it, while female offspring have no chance. This suggests that the disease is X-linked and relies on the presence of a single affected X chromosome to be expressed.

Probability Calculations

When we talk about probability calculations, we're discussing the mathematical determination of how likely an event is to occur. Probability is generally expressed as a number between 0 and 1, with 0 indicating impossibility, and 1 indicating certainty. Sometimes, probabilities are given as percentages, ranging from 0% to 100%.

To calculate the probability of a single event with multiple independent factors, such as the inheritance of a genetic disease at birth in our exercise, we multiply the probabilities of each independent factor. For instance, the overall probability of a child being affected by a sex-linked disease is the product of two probabilities: the probability of the child being male and the probability of a male inheriting the disease. Understanding how to perform these calculations is essential in genetics, as it allows us to predict the likelihood of an individual inheriting or manifesting a certain trait or genetic condition.

In our exercise, only male offspring can inherit the disease, and we're already given their probability of inheritance (50% or 0.5) and the probability of a child being male (51.3% or 0.513). Therefore, the overall probability is calculated by multiplying these two probabilities: \( 0.513 \times 0.5 = 0.2565 \) Thus, when we convert this back to a percentage, we get 25.65%, which would be the probability that a randomly chosen child will be affected by the disease.

Genetic Diseases

Genetic diseases are disorders caused by changes in an individual’s DNA. These changes can be inherited from one's parents or can occur spontaneously due to errors in DNA replication or external factors like radiation or chemicals. There's a vast range of genetic diseases, from those that are relatively common and often non-lethal (like color blindness) to rare and severe disorders (such as cystic fibrosis).

Understanding the way genetic diseases are inherited, including through sex-linked inheritance, is crucial in medical genetics and can aid in diagnosis, treatment, and genetic counseling. For individuals or couples at risk of passing on a genetic disease, knowing the probability of their children inheriting the condition is a significant concern.

Preventive measures, like genetic testing and counseling, are recommended for individuals with a family history of genetic diseases. In the case of our exercise, knowing that females cannot inherit the disease would be valuable information for prospective parents concerned about the transmission of this particular condition. Healthcare professionals use mathematical probability to inform these parents about the risks, allowing them to make more informed decisions about family planning.